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. of mdpt. 5. △ABC ≅ △EBD 5. ASA Steps 1, 4, 2 17. Given: ̶̶̶ WX ⊥ ̶̶ YZ ⊥ ̶̶̶ WZ ≅ Prove: △WZX ≅ △YXZ ̶̶ XZ, ̶̶ ZX, ̶̶ YX 18. Given: ∠S and ∠V are right angles. RT = UW. m∠T = m∠W Prove: △RST ≅ △UVW 4-6 Triangle Congruence: CPCTC (pp. 260–265) TEKS G.1.A, G.3.E, G.7.A, G.10.B E X A M P L E S ■ Given: ̶̶ JL and Prove: ∠JHG ≅ ∠LKG ̶̶ HK bisect each other. EXERCISES 19. Given: M is the midpoint ̶̶ BD. of ̶̶ BC ≅ ̶̶ DC Prove: ∠1 ≅ ∠2 Proof: Statements ̶̶ HK bisect ̶̶ 1. JL and each other. 2. ̶̶ JG ≅ ̶̶̶ HG ≅ ̶̶ LG, and ̶̶ KG. Reasons 1. Given 2. Def. of bisect 20. Given: ̶̶ RQ, ̶̶ RS ̶̶ PQ ≅ ̶̶ PS ≅ ̶̶ QS bisects ∠PQR. Prove: 3. ∠JGH ≅ ∠LGK 4. △JHG ≅ △LKG 3. Vert.  Thm. 4. SAS Steps 2, 3 5. ∠JHG ≅ ∠LKG 5. CPCTC 286 286 Chapter 4 Triangle Congruence 21. Given: H is the midpoint of L is the midpoint of ̶̶ ̶̶̶ ̶̶̶ GM ≅ KM, GJ ≅ ∠G ≅ ∠K Prove: ∠GMH ≅ ∠KJL ̶̶ KJ, ̶̶ GL. ̶̶̶ MK. �� 4-7
Introduction to Coordinate Proof (pp. 267–272) TEKS G.2.B, G.3.B, G.9.B, G.10.B E X A M P L E S EXERCISES ■ Given: ∠B is a right angle in isosceles right △ABC. E is the midpoint of ̶̶ AB ≅ D is the midpoint of ̶̶ CE ≅ Prove: Proof: Use the coordinates A(0, 2a), B(0, 0), ̶̶ CB. ̶̶ AB. ̶̶ AD ̶̶ CB and C (2a, 0). Draw ̶̶ AD and ̶̶ CE. Position each figure in the coordinate plane and give the coordinates of each vertex. 22. a right triangle with leg lengths r and s 23. a rectangle with length 2p and width p 24. a square with side length 8m For exercises 25 and 26 assign coordinates to each vertex and write a coordinate proof. 25. Given: In rectangle ABCD, E is the midpoint of ̶̶ BC, G is the ̶̶ CD, and H is the midpoint ̶̶ AB, F is the midpoint of midpoint of ̶̶ of AD. ̶̶ EF ≅ ̶̶̶ GH Prove: 26. Given: △PQR has a right ∠Q. ̶̶ PR. M is the midpoint of Prove: MP = MQ = MR 27. Show that a triangle with vertices at (3, 5), (3, 2), and (2, 5) is a right triangle. By the Midpoint Formula, E = ( D = ( 2a + + 2a _ _, 2 2 ) = (0, a) and ) = (a, 0) = (2a - 0) 2 + (0 - a) 2 By the Distance Formula, CE = √  √  4a 2 + a 2 = a √  5 √  (a - 0) 2 + (0 - 2a) 2 √ �
� = ̶̶ AD by the definition of congruence. a 2 + 4a 2 = a √  5 AD = ̶̶ CE ≅ Thus 4-8 Isosceles and Equilateral Triangles (pp. 273–279) TEKS G.2.B, G.3.C, G.10.B E X A M P L E ■ Find the value of x. m∠D + m∠E + m∠F = 180° by the Triangle Sum Theorem. m∠E = m∠F by the Isosceles Triangle Theorem. m∠D + 2 m∠E = 180° Substitution 42 + 2 (3x) = 180 Substitute the given 6x = 138 x = 23 values. Simplify. Divide both sides by 6. EXERCISES Find each value. 28. x 29. RS 30. Given: △ACD is isosceles with ∠D as the vertex angle. B is the midpoint of AB = x + 5, BC = 2x - 3, and CD = 2x + 6. ̶̶ AC. Find the perimeter of △ACD. Study Guide: Review 287 287 �� 1. Classify △ACD by its angle measures. Classify each triangle by its side lengths. 2. △ACD 3. △ABC 4. △ABD 5. While surveying the triangular plot of land shown, a surveyor finds that m∠S = 43°. The measure of ∠RTP is twice that of ∠RTS. What is m∠R? Given: △XYZ ≅ △JKL Identify the congruent corresponding parts. 6. ̶̶ JL ≅? ̶̶̶̶ 10. Given: T is the midpoint of Prove: △PTS ≅ △RTQ 7. ∠Y ≅ ̶̶ PR and? ̶̶̶̶ ̶̶ SQ. 8. ∠L ≅? ̶̶̶̶ ̶̶ YZ ≅ 9.? ̶̶̶̶ 11. The figure represents a walkway with triangular supports. Given that ∠HGK and ∠H ≅ ∠K, use AAS to
prove △HGJ ≅ △KGJ ̶̶ GJ bisects 12. Given: ̶̶ AB ≅ ̶̶ AB ⊥ ̶̶ DC ⊥ Prove: △ABC ≅ △DCB ̶̶ DC, ̶̶ AC, ̶̶ DB 13. Given: Prove: ̶̶ ̶̶ PQ ǁ SR, ∠S ≅ ∠Q ̶̶ ̶̶ QR PS ǁ 14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex. 15. Assign coordinates to each vertex and write a coordinate proof. Given: Square ABCD ̶̶ AC ≅ Prove: ̶̶ BD Find each value. 16. y 17. m∠S 18. Given: Isosceles △ABC has coordinates A (2a, 0), B (0, 2b), and C (-2a, 0). D is the midpoint of ̶̶ AC, and E is the midpoint of ̶̶ AB. Prove: △AED is isosceles. 288 288 Chapter 4 Triangle Congruence �� FOCUS ON ACT The ACT Mathematics Test is one of four tests in the ACT. You are given 60 minutes to answer 60 multiplechoice questions. The questions cover material typically taught through the end of eleventh grade. You will need to know basic formulas but nothing too difficult. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. There is no penalty for guessing on the ACT. If you are unsure of the correct answer, eliminate as many answer choices as possible and make your best guess. Make sure you have entered an answer for every question before time runs out. 1. For the figure below, which of the following must be true? 3. Which of the following best describes a triangle with vertices having coordinates (-1, 0), (0, 3), and (1, -4)? I. m∠EFG > m∠DEF II. m∠EDF = m∠EFD III. m∠DEF + m∠EDF > m∠EFG (A) I only (B) II only (C) I and II
only (D) II and III only (E) I, II, and III 2. In the figure below, △ABD ≅ △CDB, m∠A = (2x + 14) °, m∠C = (3x - 15) °, and m∠DBA = 49°. What is the measure of ∠BDA? (F) 29° (G) 49° (H) 59° (J) 72° (K) 101° (A) Equilateral (B) Isosceles (C) Right (D) Scalene (E) Equiangular 4. In the figure below, what is the value of y? (F) 49 (G) 87 (H) 93 (J) 131 (K) 136 5. In △RST, RS = 2x + 10, ST = 3x - 2, and RT = 1 __ 2 x + 28. If △RST is equiangular, what is the value of x? (A) 2 (B) 5 1 _ 3 (C) 6 (D) 12 (E) 34 College Entrance Exam Practice 289 289 �� Any Question Type: Identify Key Words and Context Clues When reading a test item, you should pay attention to key words and context clues given in the problem statement. These clues will guide you in providing a correct response. Multiple Choice What is the side length of an equilateral triangle with a perimeter of 42 3 __ 4 cm? 42 3 __ 4 cm 24 3 __ 7 cm 21 3 __ 8 cm 14 1 __ 4 cm LOOK for key words and context clues and underline them. Identify what they mean. What is the side length of an equilateral triangle with a perimeter of 42 3 __ 4 in.? equilateral triangle → → perimeter perimeter = 3 (length of one side) a triangle with three congruent sides the distance around a figure = 3 (x) = 3 (x) _ 3 42 3 _ 4 42 3 __ 4 _ 3 171 _ · 1 _ 4 3 14 1 _ 4 = x = x You find the perimeter of an equilateral triangle by multiplying the length of one side of the triangle by three. The correct choice is D because the length of the side of the equilateral triangle is 14 1 __ cm. 4 Gridded Response The vertex angle of an isosceles triangle measures (5t - 5) °, and one
of the base angles measures (t + 5) °. Find t. → isosceles triangle → vertex angle → base angles a triangle with at least two congruent sides the angle formed by the legs The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 2(measure of the base angle) + (measure of the vertex angle) = 180° 2 (t + 5 ) + (5t - 5 ) = 180 2t + 10 + 5t - 5 = 180 7t + 5 = 180 t = 25 The correct value for t is 25. 290 290 Chapter 4 Triangle Congruence �� If you do not understand what a word means, reread the sentences that contain the word and make a logical guess. 6. How will you use the abbreviation SSS to help you answer the question? Read each test item and answer the questions that follow. Item A Multiple Choice Which value of k would make △CDE isosceles. Whether a triangle is isosceles depends on what characteristics of the triangle? 2. What do 2k + 1, 3k - 7, and 5k - 6 represent in the model? 3. How will you use the definition of an isosceles triangle to find the correct value of k? Item C Multiple Choice ∠X and ∠Y are the remote interior angles of ∠YZW in △XYZ. Which of these equations must be true? 180° - m∠X = m∠YZW m∠X = m∠Y + 90° m∠X = m∠YZW - m∠Y m∠YZW = m∠YZX - m∠YXZ 7. Create a drawing that represents the situation. Label the remote interior angles. 8. What is the relationship between the remote interior angles and an exterior angle? 9. How can you manipulate the relationship given in Problem 8 to get one of the four choices? Item D Multiple Choice Which of the following is a correct classification of △FGH? Item B Gridded Response What must the value of x be in order to prove that △MNQ ≅ △PNQ by SSS? Acute Equiangular Isosceles Scalene 4. What statement are you trying to prove? 5. Explain the meaning of the symbol ≅.
10. What are the two ways by which triangles can be classified? 11. What must be true for the triangle to be classified as acute? as equiangular? 12. What must be true for the triangle to be classified as isosceles? as scalene? TAKS Tackler 291 291 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–4 Multiple Choice Use the diagram for Items 1 and 2. 6. Which conditional statement has the same truth value as its inverse? If n < 0, then n 2 > 0. If a triangle has three congruent sides, then it is an isosceles triangle. If an angle measures less than 90°, then it is an acute angle. If n is a negative integer, then n < 0. 1. Which of these congruence statements can be proved from the information given in the figure? △AEB ≅ △CED △ABD ≅ △BCA △BAC ≅ △DAC △DEC ≅ △DEA 7. On a map, an island has coordinates (3, 5), and a reef has coordinates (6, 8). If each map unit represents 1 mile, what is the distance between the island and the reef to the nearest tenth of a mile? 2. What other information is needed to prove that △CEB ≅ △AED by the HL Congruence Theorem? ̶̶̶ AD ≅ ̶̶ BE ≅ ̶̶ AB ̶̶ AE ̶̶ CB ≅ ̶̶ DE ≅ ̶̶̶ AD ̶̶ CE 3. Which biconditional statement is true? Tomorrow is Monday if and only if today is not Saturday. Next month is January if and only if this month is December. Today is a weekend day if and only if yesterday was Friday. This month had 31 days if and only if last month had 30 days. 4. What must be true if   PQ intersects   ST at more than one point? P, Q, S, and T are collinear. P, Q, S, and T are noncoplanar.  PQ and 
 ST are opposite rays.   PQ and   ST are perpendicular. 5. △ABC ≅ △DEF, EF = x 2 - 7, and BC = 4x - 2. Find the values of x. -1 and 5 -1 and 6 1 and 5 2 and 3 292 292 Chapter 4 Triangle Congruence 4.2 miles 6.0 miles 9.0 miles 15.8 miles 8. A line has an x-intercept of -8 and a y-intercept of 3. What is the equation of the line? y = -8x + = 3x - 8 9.   JK passes through points J (1, 3) and K (-3, 11). JK? Which of these lines is perpendicular to   y = -2x - = 2x - 4 10. If PQ = 2 (RS) + 4 and RS = TU + 1, which equation is true by the Substitution Property of Equality? PQ = TU + 5 PQ = TU + 6 PQ = 2 (TU) + 5 PQ = 2 (TU) + 6 11. Which of the following is NOT valid for proving that triangles are congruent? AAA ASA SAS HL �� Use this diagram for Items 12 and 13. STANDARDIZED TEST PREP Short Response 20. Given ℓ ǁ m with transversal n, explain why ∠2 and ∠3 are complementary. 12. What is the measure of ∠ACD? 40° 80° 100° 140° 13. What type of triangle is △ABC? Isosceles acute Equilateral acute Isosceles obtuse Scalene acute �� Take some time to learn the directions for filling in a grid. Check and recheck to make sure you are filling in the grid properly. You will only get credit if the ovals below the boxes are filled in correctly. To check your answer, solve the problem using a different method from the one you originally used. If you made a mistake the first time, you are unlikely to make the same mistake when you solve a different way. Gridded Response 14. △CDE ≅ △JKL. m∠E = (3
x + 4) °, and m∠L = (6x - 5) °. What is the value of x? 15. Lucy, Eduardo, Carmen, and Frank live on the same street. Eduardo’s house is halfway between Lucy’s house and Frank’s house. Lucy’s house is halfway between Carmen’s house and Frank’s house. If the distance between Eduardo’s house and Lucy’s house is 150 ft, what is the distance in feet between Carmen’s house and Eduardo’s house? 16. △JKL ≅ △XYZ, and JK = 10 - 2n. XY = 2, and YZ = n 2. Find KL. 21. ∠G and ∠H are supplementary angles. m∠G = (2x + 12) °, and m∠H = x°. a. Write an equation that can be used to determine the value of x. Solve the equation and justify each step. b. Explain why ∠H has a complement but ∠G does not. 22. A manager conjectures that for every 1000 parts a factory produces, 60 are defective. a. If the factory produces 1500 parts in one day, how many of them can be expected to be defective based on the manager’s conjecture? Explain how you found your answer. b. Use the data in the table below to show that the manager’s conjecture is false. Day Parts Defective Parts 1 2 3 4 5 1000 2000 500 1500 2500 60 150 30 90 150 23. ̶̶ BD is the perpendicular bisector of ̶̶ AC. a. What are the conclusions you can make from this statement? b. Suppose ̶̶ BD intersects is the shortest path from B to ̶̶ AC. ̶̶ AC at D. Explain why ̶̶ BD Extended Response 24. △ABC and △DEF are isosceles triangles. ̶̶ AC ≅ ̶̶ DF. m∠C = 42.5°, and m∠E = 95°. and ̶̶ BC ≅ ̶̶ EF, 17. An angle is its own supplement. What is your answer. a. What is m∠D? Explain how you determined its measure? 18. The area of a circle is 154 square inches. What is its circumference to the nearest inch
that the plane in the diagram has moved along the runway since it passed camera 1? Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List The Great Texas Balloon Race The annual Great Texas Balloon Race is one of the most exciting hot air balloon events in Texas. “Balloon Glow,” in which balloons are tethered and illuminated in an evening display, was begun in Longview, the race’s starting point, in 1980. Traditionally held in July, the race attracts balloonists who compete to fly the obstacle course the most accurately. Choose one or more strategies to solve each problem. 1. The event starts in Longview, and ends near Estes, Texas. The balloons do not fly from the start to the finish in a straight line. They follow a zigzag course to take advantage of the wind. Suppose one of the balloons leaves Longview at a bearing of N 50° E and follows the course shown. At what bearing does the balloon approach Estes? � � �� 2. The speed of the balloon depends on the current wind speed. One event in The Great Texas Balloon Race requires the balloonist to fly to a pole that is 2 mi from the starting point. The balloonist must drop a small ring around the pole, which is 20 ft tall. A second target is 1 mi from the first, a third target is another 3 mi from the second, and a final target is 5 mi farther. If the wind speed is 3.5 mi/h, how long will it take the balloonist to finish the course? Round to the nearest hundredth of an hour. 3. During the race, one of the balloons leaves Longview L, flies to X, and then flies to Y. The team discovers a problem with the balloon, so it must return directly to Longview. Does the table contain enough information to determine the return course to L? Explain to X X to Y Y to L Bearing Distance (mi) N 42° E S 59° E 3.1 2.4 Problem Solving on Location 295 295 Properties and Attributes of Triangles 5A Segments in Triangles 5-1 Perpendicular and Angle Bisectors 5-2 Bisect
ors of Triangles 5-3 Medians and Altitudes of Triangles Lab Special Points in Triangles 5-4 The Triangle Midsegment Theorem 5B Relationships in Triangles Lab Explore Triangle Inequalities 5-5 Indirect Proof and Inequalities in One Triangle 5-6 Inequalities in Two Triangles Lab Hands-on Proof of the Pythagorean Theorem 5-7 The Pythagorean Theorem 5-8 Applying Special Right Triangles Lab Graph Irrational Numbers KEYWORD: MG7 ChProj The Broken Obelisk in Houston was dedicated in 1971 as a memorial to Martin Luther King Jr. 296 296 Chapter 5 Vocabulary Match each term on the left with a definition on the right. 1. angle bisector A. the side opposite the right angle in a right triangle 2. conclusion 3. hypotenuse 4. leg of a right triangle 5. perpendicular bisector of a segment B. a line that is perpendicular to a segment at its midpoint C. the phrase following the word then in a conditional statement D. one of the two sides that form the right angle in a right triangle E. a line or ray that divides an angle into two congruent angles F. the phrase following the word if in a conditional statement Classify Triangles Tell whether each triangle is acute, right, or obtuse. 7. 6. 8. 9. Squares and Square Roots Simplify each expression. 10. 8 2 11. (-12)2 Simplify Radical Expressions Simplify each expression. 14. √9 + 16 15. √ 100 - 36 12. √49 13. - √36 16. √81_ 25 17. √ 2 2 Solve and Graph Inequalities Solve each inequality. Graph the solutions on a number line. 19. -4 ≤ w - 7 18. d + 5 < 1 20. -3s ≥ 6 21. -2 > m_ 10 Logical Reasoning Draw a conclusion from each set of true statements. 22. If two lines intersect, then they are not parallel. Lines ℓ and m intersect at P. 23. If M is the midpoint of ̶̶ AB, then AM = MB. If AM = MB, then AM = 1 __ 2 AB and MB =
1 __ 2 AB. Properties and Attributes of Triangles 297 297 �� Key Vocabulary/Vocabulario altitude of a triangle altura de un triángulo centroid of a triangle centroide de un triángulo circumcenter of a triangle circuncentro de un triángulo concurrent equidistant concurrente equidistante incenter of a triangle incentro de un triángulo median of a triangle mediana de un triángulo midsegment of a triangle segmento medio de un triángulo orthocenter of a triangle orthocentro de un triángulo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. In Latin, co means “together with,” and currere means “to run.” How can you use these meanings to understand what concurrent lines are? 2. The endpoints of a midsegment of a triangle are on two sides of the triangle. Where on the sides do you think the endpoints are located? 3. The strip of concrete or grass in the middle of some roadways is called the median. What do you think the term median of a triangle means? 4. Think of the everyday meaning of altitude. What do you think the altitude of a triangle is? Geometry TEKS Les. 5-1 Les. 5-2 Les. 5-3 5-3 Tech. Lab 5-5 Geo. Lab Les. 5-4 Les. 5-5 Les. 5-6 5-7 Geo. Lab Les. 5-7 Les. 5-8 G.2.A Geometric structure* use constructions to ★ ★ ★ ★ explore attributes of geometric figures and to make conjectures... G.3.B Geometric structure* construct and justify ★ ★ ★ ★ ★ ★ ★ 5-8 Geo. Lab ★ statements about geometric figures and their properties G.5.D Geometric patterns* identify and apply patterns from right triangles... including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples G.7.B Dimensionality and the geometry of location... investigate geometric relationships, including... special segments of triangles... G.8.C Congruence and the geometry of size derive, extend
, and use the Pythagorean Theorem G.9.B Congruence and the geometry of size* formulate and test conjectures about... polygons and their component parts... G.11.C Similarity and the geometry of shape* develop, apply, and justify triangle similarity relationships, such as... Pythagorean triples... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 298 298 Chapter 5 Reading Strategy: Learn Math Vocabulary Mathematics has a vocabulary all its own. To learn and remember new vocabulary words, use the following study strategies. • Try to figure out the meaning of a new word based on its context. • Use a dictionary to look up the root word or prefix. • Relate the new word to familiar everyday words. Once you know what a word means, write its definition in your own words. Term Study Notes Definition Polygon The prefix poly means “many” or “several.” Bisect Slope Intersection The prefix bi means “two.” Think of a ski slope. The root word intersect means “to overlap.” Think of the intersection of two roads. A closed plane figure formed by three or more line segments Cuts or divides something into two equal parts The measure of the steepness of a line The set of points that two or more lines have in common Try This Complete the table below. Term Study Notes Definition 1. Trinomial 2. Equiangular triangle 3. Perimeter 4. Deductive reasoning Use the given prefix and its meanings to write a definition for each vocabulary word. 5. circum (about, around); circumference 6. co (with, together); coplanar 7. trans (across, beyond, through); translation Properties and Attributes of Triangles 299 299 5-1 Perpendicular and Angle Bisectors TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.E, G.7.A, G.7.B, G.7.C, G.10.B Objectives Prove and apply theorems about perpendicular bisectors. Prove and apply theorems about angle bisectors. Vocabulary equidistant locus Who uses this? The suspension and steering lines of a parachute keep the sky diver centered under the parachute. (See Example 3.) When a point is the same distance from
two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points. Theorems Distance and Perpendicular Bisectors THEOREM HYPOTHESIS CONCLUSION 5-1-1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 5-1-2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. XA = XB ̶̶ XY ⊥ ̶̶ YA ≅ ̶̶ AB ̶̶ YB ̶̶ XY ⊥ ̶̶ YA ≅ ̶̶ AB ̶̶ YB XA = XB You will prove Theorem 5-1-2 in Exercise 30. PROOF PROOF Perpendicular Bisector Theorem Given: ℓ is the perpendicular bisector of Prove: XA = XB ̶̶ AB. The word locus comes from the Latin word for location. The plural of locus is loci, which is pronounced LOW-sigh. Proof: ̶̶ AB, ℓ ⊥ ̶̶ AB and Y is the midpoint ̶̶ AB. By the definition of perpendicular, ∠AYX and ∠BYX are right ̶̶ BY. Since ℓ is the perpendicular bisector of of angles and ∠AYX ≅ ∠BYX. By the definition of midpoint, By the Reflexive Property of Congruence, by SAS, and of congruent segments. ̶̶ XY. So △AYX ≅ △BYX ̶̶ XB by CPCTC. Therefore XA = XB by the definition ̶̶ XA ≅ ̶̶ XY ≅ ̶̶ AY ≅ A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. 300 300 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Applying the Perpendicular Bisector The
orem and Its Converse Find each measure. A YW YW = XW YW = 7.3 B BC ⊥ Bisector Thm. Substitute 7.3 for XW. ̶̶ BC, ℓ is the perpendicular Since AB = AC and ℓ ⊥ ̶̶ BC by the Converse of the bisector of Perpendicular Bisector Theorem. BC = 2CD BC = 2 (16) = 32 Def. of seg. bisector Substitute 16 for CD. C PR PR = RQ 2n + 9 = 7n - 18 9 = 5n - 18 27 = 5n 5.4 = n ⊥ Bisector Thm. Substitute the given values. Subtract 2n from both sides. Add 18 to both sides. Divide both sides by 5. So PR = 2 (5.4) + 9 = 19.8. Find each measure. 1a. Given that line ℓ is the perpendicular ̶̶ DE and EG = 14.6, find DG. bisector of 1b. Given that DE = 20.8, DG = 36.4, and EG = 36.4, find EF. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. Theorems Distance and Angle Bisectors THEOREM HYPOTHESIS CONCLUSION 5-1-3 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. 5-1-4 Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. ∠APC ≅ ∠BPC AC = BC ∠APC ≅ ∠BPC AC = BC You will prove these theorems in Exercises 31 and 40. 5- 1 Perpendicular and Angle Bisectors 301 301 �� Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle. E X A M P L E 2 Applying the Angle Bisector Theorems Find each measure. A LM LM = JM LM = 12
.8 ∠ Bisector Thm. Substitute 12.8 for JM. B m∠ABD, given that m∠ABC = 112° ̶̶ BA, and ̶̶ AD ⊥  BD bisects ∠ABC Since AD = DC, ̶̶ ̶̶ DC ⊥ BC, by the Converse of the Angle Bisector Theorem. m∠ABD = 1 _ 2 m∠ABD = 1 _ (112°) = 56° Substitute 112° for m∠ABC. 2 Def. of ∠ bisector m∠ABC C m∠TSU ̶̶ UT ⊥ ̶̶ SR, and ̶̶ ̶̶ Since RU = UT, ST, RU ⊥  SU bisects ∠RST by the Converse of the Angle Bisector Theorem. m∠RSU = m∠TSU 6z + 14 = 5z + 23 z + 14 = 23 Def. of ∠ bisector Substitute the given values. Subtract 5z from both sides. z = 9 Subtract 14 from both sides. ⎤ ⎦ ⎡ ⎣ ° = 68°. 5 (9) + 23 So m∠TSU = Find each measure. 2a. Given that  YW bisects ∠XYZ and WZ = 3.05, find WX. 2b. Given that m∠WYZ = 63°, XW = 5.7, and ZW = 5.7, find m∠XYZ. E X A M P L E 3 Parachute Application Each pair of suspension lines on a parachute are the same length and are equally spaced from the center of the chute. How do these lines keep the sky diver centered under the parachute? ̶̶ PQ ≅ ̶̶ RQ. So Q is It is given that on the perpendicular bisector ̶̶ PR by the Converse of the of Perpendicular Bisector Theorem. Since S is the midpoint of ̶̶ QS is the perpendicular bisector ̶̶ PR. Therefore the sky diver of remains centered under the chute. ̶̶ PR, 302 302 Chapter 5 Properties and Attributes of Triangles ��
�� 3. S is equidistant from each pair of suspension lines. What can you conclude about  QS? E X A M P L E 4 Writing Equations of Bisectors in the Coordinate Plane Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints A (-1, 6) and B (3, 4). ̶̶ AB. Step 1 Graph The perpendicular bisector of perpendicular to ̶̶ AB is ̶̶ AB at its midpoint. ̶̶ AB. Step 2 Find the midpoint of AB = ( 6 + 4 -1 + 3 _ _, 2 2 mdpt. of ̶̶ ) = (1, 5) Midpoint formula Step 3 Find the slope of the perpendicular bisector. slope = slope of ̶̶ AB = 4 - 6 _ 3 - (-1) Slope formula = -2 _ = - 1 _ 4 2 Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is 2. Step 4 Use point-slope form to write an equation. The perpendicular bisector of ̶̶ AB has slope 2 and passes through (1, 5). y - y 1 = m (x - 1 ) Point-slope form Substitute 5 for y 1, 2 for m, and 1 for x 1. 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P (5, 2) and Q (1, -4). THINK AND DISCUSS ̶̶ PQ? Is it a perpendicular 1. Is line ℓ a bisector of bisector of ̶̶ PQ? Explain. 2. Suppose that M is in the interior of ∠JKL and MJ = ML. Can you conclude that is the bisector of ∠JKL? Explain.  KM 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the theorem or its converse in your own words. 5- 1 Perpendicular and Angle Bisectors 303 303 �� 5-1 Exercises Exercises KEYWORD: MG7 5-1 KEYWORD: MG7 Parent GUIDED PRACTICE
1. Vocabulary A from the endpoints of a segment. (perpendicular bisector or angle bisector)? is the locus of all points in a plane that are equidistant ̶̶̶̶ Use the diagram for Exercises 2–4. p. 301 2. Given that PS = 53.4, QT = 47.7, and QS = 53.4, find PQ. 3. Given that m is the perpendicular bisector ̶̶ PQ and SQ = 25.9, find SP. of 4. Given that m is the perpendicular bisector ̶̶ PQ, PS = 4a, and QS = 2a + 26, find QS. of Use the diagram for Exercises 5–7. p. 302 5. Given that  BD bisects ∠ABC and CD = 21.9, find AD. 6. Given that AD = 61, CD = 61, and m∠ABC = 48°, find m∠CBD. 7. Given that DA = DC, m∠DBC = (10y + 3) °, and m∠DBA = (8y + 10) °, find m∠DBC. 302 8. Carpentry For a king post truss to be constructed correctly, P must lie on the bisector of ∠JLN. How can braces and the proper location? ̶̶ PK ̶̶̶ PM be used to ensure that P is in. 303 Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. M (-5, 4), N (1, -2) 10. U (2, -6), V (4, 0) 11. J (-7, 5), K (1, -1) Independent Practice Use the diagram for Exercises 12–14. PRACTICE AND PROBLEM SOLVING For See Exercises Example 12–14 15–17 18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 12. Given that line t is the perpendicular bisector ̶̶ JK and GK = 8.25, find GJ. of 13. Given that line t is the perpendicular bisector ̶̶ JK, JG = x +
12, and KG = 3x - 17, find KG. of 14. Given that GJ = 70.2, JH = 26.5, and GK = 70.2, find JK. Use the diagram for Exercises 15–17. 15. Given that m∠RSQ = m∠TSQ and TQ = 1.3, find RQ. 16. Given that m∠RSQ = 58°, RQ = 49, and TQ = 49, find m∠RST. 17. Given that RQ = TQ, m∠QSR = (9a + 48) °, and m∠QST = (6a + 50) °, find m∠QST. 304 304 Chapter 5 Properties and Attributes of Triangles ��ge07sec05l01003aa1st pass4/4/5cmurphyLMNKJP�� 18. City Planning The planners for a new section of the city want every location on Main Street to be equidistant from Elm Street and Grove Street. How can the planners ensure that this is the case? Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 19. E (-4, -7), F (0, 1) 20. X (-7, 5), Y (-1, -1) ̶̶ ST. 22. ̶̶ PQ is the perpendicular bisector of Find the values of m and n. 21. M (-3, -1), N (7, -5) Shuffleboard One of the first recorded shuffleboard games was played in England in 1532. In this game, Henry VIII supposedly lost £9 to Lord William. Shuffleboard Use the diagram of a shuffleboard and the following information to find each length in Exercises 23–28. ̶̶ KZ is the perpendicular bisector of ̶̶̶ HM, and ̶̶̶ GN, ̶̶ JL. 23. JK 24. GN 25. ML 27. JL 26. HY 29. Multi-Step The endpoints of 28. NM ̶̶ AB are A (-2, 1) and B (4, -3). Find the coordinates of a point C other than the midpoint of you know it is on the perpendicular bisector? � ��
�� ̶̶ AB that is on the perpendicular bisector of ̶̶ AB. How do 30. Write a paragraph proof of the Converse of the Perpendicular Bisector Theorem. Given: AX = BX Prove: X is on the perpendicular bisector of ̶̶ AB. Plan: Draw ℓ perpendicular to ̶̶ △AYX ≅ △BYX and thus AY ≅ the perpendicular bisector of ̶̶ AB through X. Show that ̶̶ BY. By definition, ℓ is ̶̶ AB. 31. Write a two-column proof of the Angle Bisector Theorem. ̶̶ SQ ⊥  PS bisects ∠QPR. ̶̶ SR ⊥  PQ,  PR Given: Prove: SQ = SR Plan: Use the definitions of angle bisector and perpendicular to identify two pairs of congruent angles. Show that △PQS ≅ △PRS and thus ̶̶ SQ ≅ ̶̶ SR. 32. Critical Thinking In the Converse of the Angle Bisector Theorem, why is it important to say that the point must be in the interior of the angle? 33. This problem will prepare you for the Multi-Step TAKS Prep on page 328. A music company has stores in Abby (-3, -2) and Cardenas (3, 6). Each unit in the coordinate plane represents 1 mile. a. The company president wants to build a warehouse that is equidistant from the two stores. Write an equation that describes the possible locations. b. A straight road connects Abby and Cardenas. The warehouse will be located exactly 4 miles from the road. How many locations are possible? c. To the nearest tenth of a mile, how far will the warehouse be from each store? 5- 1 Perpendicular and Angle Bisectors 305 305 Elm StreetMain StreetGrove StreetHolt, Rinehart & WinstonGeometry 2007ge07se_c05l01004a 1st proofCity Planning map�� 34. Write About It How is the construction of the perpendicular bisector of a segment related to the Converse of the Perpendicular Bisector Theorem? 35.
If   JK is perpendicular to JX = KY ̶̶ XY at its midpoint M, which statement is true? JX = KX JM = KM JX = JY 36. What information is needed to conclude that m∠DEF = m∠DEG m∠FEG = m∠DEF  EF is the bisector of ∠DEG? m∠GED = m∠GEF m∠DEF = m∠EFG 37. Short Response The city wants to build a visitor center in the park so that it is equidistant from Park Street and Washington Avenue. They also want the visitor center to be equidistant from the museum and the library. Find the point V where the visitor center should be built. Explain your answer. CHALLENGE AND EXTEND 38. Consider the points P (2, 0), A (-4, 2), B (0, -6), and C (6, -3). a. Show that P is on the bisector of ∠ABC. b. Write an equation of the line that contains the bisector of ∠ABC. 39. Find the locus of points that are equidistant from the x-axis and y-axis. 40. Write a two-column proof of the Converse of the Angle Bisector Theorem. Given: Prove: ̶̶ ̶̶ VZ ⊥  YX, VX ⊥  YV bisects ∠XYZ.  YZ, VX = VZ 41. Write a paragraph proof. Given: ̶̶ KN is the perpendicular bisector of ̶̶ LN is the perpendicular bisector of ̶̶ JR ≅ Prove: ∠JKM ≅ ∠MLJ ̶̶̶ MT ̶̶ JL. ̶̶̶ KM. SPIRAL REVIEW 42. Lyn bought a sweater for $16.95. The change c that she received can be described by c = t - 16.95, where t is the amount of money Lyn gave the cashier. What is the dependent variable? (Previous course) For the points R (-4, 2), S (1, 4), T
(3, -1), and V (-7, -5), determine whether the lines are parallel, perpendicular, or neither. (Lesson 3-5) 43.   RS and   VT 44.   RV and   ST 45.   RT and   VR Write the equation of each line in slope-intercept form. (Lesson 3-6) 46. the line through the points (1, -1) and (2, -9) 47. the line with slope -0.5 through (10, -15) 48. the line with x-intercept -4 and y-intercept 5 306 306 Chapter 5 Properties and Attributes of Triangles ge07sec05l01006a3rd Pass2/24/05C MurphyMuseumPark StreetWashington AvenueLibrary�� 5-2 Bisectors of Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.A, G.7.B Objectives Prove and apply properties of perpendicular bisectors of a triangle. Prove and apply properties of angle bisectors of a triangle. Vocabulary concurrent point of concurrency circumcenter of a triangle circumscribed incenter of a triangle inscribed The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex. Who uses this? An event planner can use perpendicular bisectors of triangles to find the best location for a fireworks display. (See Example 4.) Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property. Construction Circumcenter of a Triangle    Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point
of concurrency is the circumcenter of the triangle. Theorem 5-2-1 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC The circumcenter can be inside the triangle, outside the triangle, or on the triangle. 5- 2 Bisectors of Triangles 307 307 �� The circumcenter of △ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. PROOF PROOF Circumcenter Theorem Given: Lines ℓ, m, and n are the perpendicular ̶̶ ̶̶ AC, respectively. BC, and ̶̶ AB, bisectors of Prove: PA = PB = PC Proof: P is the circumcenter of △ABC. Since P lies on the perpendicular bisector of by the Perpendicular Bisector Theorem. Similarly, P also lies on the perpendicular bisector of by the Transitive Property of Equality. ̶̶ BC, so PB = PC. Therefore PA = PB = PC ̶̶ AB, PA = PB E X A M P L E 1 Using Properties of Perpendicular Bisectors ̶̶ LZ, and ̶̶ KZ, of △GHJ. Find HZ. ̶̶ MZ are the perpendicular bisectors Z is the circumcenter of △GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of △GHJ. HZ = GZ HZ = 19.9 Circumcenter Thm. Substitute 19.9 for GZ. Use the diagram above. Find each length. 1a. GM 1b. GK 1c. JZ E X A M P L E 2 Finding the Circumcenter of a Triangle Find the circumcenter of △RSO with vertices R (-6, 0), S (0, 4), and O (0, 0). Step 1 Graph the triangle. Step 2 Find equations for two perpendicular bisectors. Since two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of x = -3, and the perpendicular bisector of ̶̶ OS is y = 2. ̶̶ RO is Step 3
Find the intersection of the two equations. The lines x = -3 and y = 2 intersect at (-3, 2 ), the circumcenter of △RSO. 308 308 Chapter 5 Properties and Attributes of Triangles �� 2. Find the circumcenter of △GOH with vertices G (0, -9), O (0, 0), and H (8, 0). A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle. Theorem 5-2-2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ You will prove Theorem 5-2-2 in Exercise 35. Unlike the circumcenter, the incenter is always inside the triangle. The distance between a point and a line is the length of the perpendicular segment from the point to the line. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point. E X A M P L E 3 Using Properties of Angle Bisectors ̶̶ KV are angle bisectors of △JKL. ̶̶ JV and Find each measure. A the distance from V to ̶̶ KL V is the incenter of △JKL. By the Incenter Theorem, V is equidistant from the sides of △JKL. The distance from V to So the distance from V to ̶̶ JK is 7.3. ̶̶ KL is also 7.3. B m∠VKL m∠KJL = 2m∠VJL m∠KJL = 2 (19°) = 38° m∠KJL + m∠JLK + m∠JKL = 180° 38 + 106 + m∠JKL = 180 m∠JKL = 36° m∠JKL m∠VKL = 1 _ 2 m∠VKL = 1 _ (36°) = 18° 2 ̶̶ JV is the bisector of ∠KJL. Substitute 19° for m�
�VJL. △ Sum Thm. Substitute the given values. Subtract 144° from both sides. ̶̶ KV is the bisector of ∠JKL. Substitute 36° for m∠JKL. 5- 2 Bisectors of Triangles 309 309 �� ̶̶ RX are angle bisectors ̶̶ QX and of △PQR. Find each measure. ̶̶ PQ 3a. the distance from X to 3b. m∠PQX E X A M P L E 4 Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing locations A, B, and C on the map. Justify your sketch. Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. 4. A city plans to build a � � � � firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch. � � � � � � � � �� THINK AND DISCUSS 1. Sketch three lines that are concurrent. 2. P and Q are the circumcenter and incenter of △RST, but not necessarily in that order. Which point is the circumcenter? Which point is the incenter? Explain how you can tell without constructing any of the bisectors. 3. GET ORGANIZED Copy and complete the graphic organizer. Fill in the blanks to make each statement true. 310 310 Chapter 5 Properties and Attributes of Triangles BAC38520print-ready file 7/12/05ge07ts_c05l02005aGeometry SE 2007 Texasmap of OdessaHolt Rinehart WinstonKaren Minot(415)883
-6560�� 5-2 Exercises Exercises KEYWORD: MG7 5-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain why lines ℓ, m, and n are NOT concurrent. 2. A circle that contains all the vertices of a polygon is. 308? the polygon. (circumscribed about or inscribed in) ̶̶̶̶ ̶̶ TN, and ̶̶ VN are the perpendicular bisectors ̶̶ SN, of △PQR. Find each length. 3. NR 5. TR 4. RV 6. QN Multi-Step Find the circumcenter of a triangle with the given vertices. p. 308 7. O (0, 0), K (0, 12), L (4, 0. 309 8. A (-7, 0), O (0, 0), B (0, -10) ̶̶ CF and Find each measure. ̶̶ EF are angle bisectors of △CDE. 9. the distance from F to ̶̶ CD 10. m∠FED 11. Design The designer of the p. 310 Newtown High School pennant wants the circle around the bear emblem to be as large as possible. Draw a sketch to show where the center of the circle should be located. Justify your sketch. Independent Practice For See Exercises Example 12–15 16–17 18–19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING ̶̶ DY, of △ABC. Find each length. ̶̶ FY are the perpendicular bisectors ̶̶ EY, and 12. CF 14. DB 13. YC 15. AY Multi-Step Find the circumcenter of a triangle with the given vertices. 16. M (-5, 0), N (0, 14), O (0, 0)
17. O (0, 0), V (0, 19), W (-3, 0) ̶̶ SJ are angle bisectors of △RST. ̶̶ TJ and Find each measure. 18. the distance from J to ̶̶ RS 19. m∠RTJ 5- 2 Bisectors of Triangles 311 311 �� 20. Business A company repairs photocopiers in Harbury, Gaspar, and Knowlton. Draw a sketch to show where the company should locate its office so that it is the same distance from each city. Justify your sketch. 21. Critical Thinking If M is the incenter of △JKL, explain why ∠JML cannot be a right angle. Tell whether each segment lies on a perpendicular bisector, an angle bisector, or neither. Justify your answer. 22. 25. ̶̶ AE ̶̶ CR 23. 26. ̶̶̶ DG ̶̶ FR 24. 27. ̶̶ BG ̶̶ DR Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 28. The angle bisectors of a triangle intersect at a point outside the triangle. 29. An angle bisector of a triangle bisects the opposite side. 30. A perpendicular bisector of a triangle passes through the opposite vertex. 31. The incenter of a right triangle is on the triangle. 32. The circumcenter of a scalene triangle is inside the triangle. Algebra Find the circumcenter of the triangle with the given vertices. 33. O (0, 0), A (4, 8), B (8, 0) 34. O (0, 0), Y (0, 12), Z (6, 6) 35. Complete this proof of the Incenter Theorem by filling in the blanks. Given:  AP, ̶̶ PX ⊥ Prove: PX = PY = PZ  BP, and ̶̶ ̶̶ PY ⊥ AC, ̶̶ AB, ̶̶ PZ ⊥ ̶̶ BC  CP bisect ∠A, ∠B, and ∠C, respectively. Proof:
Let P be the incenter of △ABC. Since P lies on the bisector of ∠A, PX = PY by a. Similarly, P also lies on b. Therefore c.?. ̶̶̶̶?, so PY = PZ. ̶̶̶̶? by the Transitive Property of Equality. ̶̶̶̶ 36. Prove that the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Given:   QS bisects ∠PQR. Prove:   QS is the perpendicular bisector of ̶̶ PR. ̶̶ PQ ≅ ̶̶ RQ Plan: Show that △PQS ≅ △RQS. Then use CPCTC to show that S is the midpoint of ̶̶ PR and that   QS ⊥ ̶̶ PR. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 328. A music company has stores at A (0, 0), B (8, 0), and C (4, 3), where each unit of the coordinate plane represents one mile. a. A new store will be built so that it is equidistant from the three existing stores. Find the coordinates of the new store’s location. b. Where will the new store be located in relation to △ABC? c. To the nearest tenth of a mile, how far will the new store be from each of the existing stores? 312 312 Chapter 5 Properties and Attributes of Triangles HarburyGasparKnowltonHarburyGasparKnowltonHolt, Rinehart & WinstonGeometry 2007ge07se_c05l02008a 1st pass3/9/5C MurphyBisector map�� 38. Write About It How are the inscribed circle and the circumscribed circle of a triangle alike? How are they different? 39. Construction Draw a large scalene acute triangle. a. Construct the angle bisectors to find the incenter. Inscribe a circle in the triangle. b. Construct the perpendicular bisectors to find the circumcenter. Circumscribe a circle around the triangle. 40. P is the incenter of △ABC.
Which must be true? PA = PB PX = PY YA = YB AX = BZ 41. Lines r, s, and t are concurrent. The equation of line r is x = 5, and the equation of line s is y = -2. Which could be the equation of line t 42. Gridded Response Lines a, b, and c are the perpendicular bisectors of △KLM. Find LN. CHALLENGE AND EXTEND 43. Use the right triangle with the given coordinates. a. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. b. Make a conjecture about the circumcenter of a right triangle. 44. Design A trefoil is created by constructing a circle at each vertex of an equilateral triangle. The radius of each circle equals the distance from each vertex to the circumcenter of the triangle. If the distance from one vertex to the circumcenter is 14 cm, what is the distance AB across the trefoil? Design The trefoil shape, as seen in this stained glass window, has been used in design for centuries. SPIRAL REVIEW Solve each proportion. (Previous course) 45. t _ 26 46. 2.5 _ 1.75 = 10 _ 65 = 6 _ x 47. 420 _ y = 7 _ 2 Find each angle measure. (Lesson 2-6) 48. m∠BFE 49. m∠BFC 50. m∠CFE Determine whether each point is on the perpendicular bisector of the segment with endpoints S (0, 8) and T (4, 0). (Lesson 5-1) 51. X (0, 3) 53. Z (-8, -2) 52. Y (-4, 1) 5- 2 Bisectors of Triangles 313 313 �� 5-3 Medians and Altitudes of Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.A, G.7.B, G.7.C Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle. Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a
triangle Who uses this? Sculptors who create mobiles of moving objects can use centers of gravity to balance the objects. (See Example 2.) A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent, as shown in the construction below. Construction Centroid of a Triangle    Draw △ABC. Construct the ̶̶ midpoints of BC, and Label the midpoints of the sides X, Y, and Z, respectively. ̶̶ AB, ̶̶ AC. ̶̶ AY, ̶̶ Draw BZ, and the three medians of △ABC. ̶̶ CX. These are Label the point where and ̶̶ CX intersect as P. ̶̶ AY, ̶̶ BZ, The point of concurrency of the medians of a triangle is the centroid of the triangle. The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. Theorem 5-3-1 Centroid Theorem The centroid of a triangle is located 2 __ of the distance 3 from each vertex to the midpoint of the opposite side. AP = 2 _ 3 AY BP = 2 _ 3 BZ CP = 2 _ 3 CX 314 314 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Centroid to Find Segment Lengths In △ABC, AF = 9, and GE = 2.4. Find each length. A AG AF AG = 2 _ 3 AG = 2 _ (9) 3 AG = 6 B CE CG = 2 _ 3 CE Centroid Thm. Substitute 9 for AF. Simplify. Centroid Thm. CG + GE = CE 2 _ CE + GE = CE 3 GE = 1 _ 3 2.4 = 1 _ 3 CE CE Seg. Add. Post. Substitute 2 _ 3 Subtract 2 _ 3 CE for CG. CE from both sides. Substitute 2.4 for GE. 7.2 = CE Multiply both sides by 3. In △JKL, ZW = 7, and LX = 8.1. Find each length. 1a. KW 1b. LZ E X
A M P L E 2 Problem-Solving Application The diagram shows the plan for a triangular piece of a mobile. Where should the sculptor attach the support so that the triangle is balanced? Understand the Problem The answer will be the coordinates of the centroid of △PQR. The important information is the location of the vertices, P (3, 0), Q (0, 8), and R (6, 4). Make a Plan The centroid of the triangle is the point of intersection of the three medians. So write the equations for two medians and find their point of intersection. Solve Let M be the midpoint of ̶̶ QR and N be the midpoint of ̶̶ QP. 3, 61.5, 4) ̶̶̶ PM is vertical. Its equation is x = 3. Its equation is y = 4. The coordinates of the centroid are S (3, 4). ̶̶ RN is horizontal. 5- 3 Medians and Altitudes of Triangles 315 315 ��123 Look Back Let L be the midpoint of intersects x = 3 at S (3, 4). ̶̶ PR. The equation for ̶̶ QL is y = - 4 _ 3 x + 8, which The height of a triangle is the length of an altitude. 2. Find the average of the x-coordinates and the average of the y-coordinates of the vertices of △PQR. Make a conjecture about the centroid of a triangle. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. ̶̶ QY is inside the triangle, but ̶̶ SZ are not. Notice that the lines containing In △QRS, altitude and the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle. ̶̶ RX E X A M P L E 3 Finding the Orthocenter Find the orthocenter of △JKL with vertices J (-4, 2), K (-2, 6), and L (2, 2). Step 1 Graph the triangle. Step 2 Find an equation of the line containing the altitude from K to ̶̶ JL. JL is horizontal, the altitude is
Since   vertical. The line containing it must pass through K (-2, 6), so the equation of the line is x = -2. Step 3 Find an equation of the line containing the altitude from J to ̶̶ KL. slope of   KL = 2 - 6 _ = -1 2 - (-2) The slope of a line perpendicular to   KL is 1. This line must pass through J (-4, 2). y - y 1 = m ( - (-4 Point-slope form Substitute 2 for y 1, 1 for m, and -4 for x 1. Distribute 1. Add 2 to both sides. Step 4 Solve the system to find the coordinates of the orthocenter. ⎧ x = -2 + 6 = 4 Substitute -2 for x. The coordinates of the orthocenter are (-2, 4). 3. Show that the altitude to ̶̶ JK passes through the orthocenter of △JKL. 316 316 Chapter 5 Properties and Attributes of Triangles 4�� THINK AND DISCUSS 1. Draw a triangle in which a median and an altitude are the same segment. What type of triangle is it? 2. Draw a triangle in which an altitude is also a side of the triangle. What type of triangle is it? 3. The centroid of a triangle divides each median into two segments. What is the ratio of the two lengths of each median? 4. GET ORGANIZED Copy and complete the graphic organizer. Fill in the blanks to make each statement true. 5-3 Exercises Exercises KEYWORD: MG7 5-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question.? of a triangle is located 2 __ 3 of the distance from each vertex to the ̶̶̶̶ 1. The midpoint of the opposite side. (centroid or orthocenter) 2. The? of a triangle is perpendicular to the line containing a side. ̶̶̶̶ (altitude or median VX = 204, and RW = 104. Find each length. p. 315 3. VW 5. RY 4. WX 6. WY. Design The diagram shows a plan for p. 315 a
piece of a mobile. A chain will hang from the centroid of the triangle. At what coordinates should the artist attach the chain Multi-Step Find the orthocenter of a triangle with the given vertices. p. 316 8. K (2, -2), L (4, 6), M (8, -2) 9. U (-4, -9), V (-4, 6), W (5, -3) 10. P (-5, 8), Q (4, 5), R (-2, 5) 11. C (-1, -3), D (-1, 2), E (9, 2) 5- 3 Medians and Altitudes of Triangles 317 317 �� Independent Practice For See Exercises Example 12–15 16 17–20 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING PA = 2.9, and HC = 10.8. Find each length. 12. PC 14. JA 13. HP 15. JP 16. Design In the plan for a table, the triangular top has coordinates (0, 10), (4, 0), and (8, 14). The tabletop will rest on a single support placed beneath it. Where should the support be attached so that the table is balanced? Multi-Step Find the orthocenter of a triangle with the given vertices. 17. X (-2, -2), Y (6, 10), Z (6, -6) 18. G (-2, 5), H (6, 5), J (4, -1) 19. R (-8, 9), S (-2, 9), T (-2, 1) 20. A (4, -3), B (8, 5), C (8, -8) Find each measure. 21. GL 23. HL 22. PL 24. GJ 25. perimeter of △GHJ 26. area of △GHJ Algebra Find the centroid of a triangle with the given vertices. 27. A (0, -4), B
(14, 6), C (16, -8) 28. X (8, -1), Y (2, 7), Z (5, -3) Find each length. 29. PZ 31. QZ 30. PX 32. YZ Math History 33. Critical Thinking Draw an isosceles triangle and its line of symmetry. What are four other names for this segment? Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 34. A median of a triangle bisects one of the angles. 35. If one altitude of a triangle is in the triangle’s exterior, then a second altitude is also in the triangle’s exterior. 36. The centroid of a triangle lies in its exterior. 37. In an isosceles triangle, the altitude and median from the vertex angle are the same line as the bisector of the vertex angle. 38. Write a two-column proof. ̶̶ PS and Given: Prove: △PQR is an isosceles triangle. ̶̶ RT are medians of △PQR. ̶̶ PS ≅ ̶̶ RT In 1678, Giovanni Ceva published his famous theorem that states the conditions necessary for three Cevians (segments from a vertex of a triangle to the opposite side) to be concurrent. The medians and altitudes of a triangle meet these conditions. Plan: Show that △PTR ≅ △RSP and use CPCTC to conclude that ∠QPR ≅ ∠QRP. 39. Write About It Draw a large triangle on a sheet of paper and cut it out. Find the centroid by paper folding. Try to balance the shape on the tip of your pencil at a point other than the centroid. Now try to balance the shape at its centroid. Explain why the centroid is also called the center of gravity. 318 318 Chapter 5 Properties and Attributes of Triangles �� 40. This problem will prepare you for the Multi-Step TAKS Prep on page 328. The towns of Davis, El Monte, and Fairview have the coordinates shown in the table, where each unit of the coordinate plane represents one mile. A music company has stores in each city and a distribution warehouse at the centroid of △DEF. a. What are the coordinates of the warehouse? b.
Find the distance from the warehouse to the City Location Davis El Monte Fairview D (0, 0) E (0, 8) F (8, 0) Davis store. Round your answer to the nearest tenth of a mile. c. A straight road connects El Monte and Fairview. What is the distance from the warehouse to the road? 41. ̶̶ RV, and ̶̶ QT, NOT necessarily true? ̶̶̶ SW are medians of △QRS. Which statement is QP = 2 _ 3 RP = 2PV QT RT = ST QT = SW 42. Suppose that the orthocenter of a triangle lies outside the triangle. Which points of concurrency are inside the triangle? I. incenter II. circumcenter III. centroid I and II only I and III only II and III only I, II, and III 43. In the diagram, which of the following correctly describes ̶̶ LN? Altitude Median Angle bisector Perpendicular bisector CHALLENGE AND EXTEND 44. Draw an equilateral triangle. a. Explain why the perpendicular bisector of any side contains the vertex opposite that side. b. Explain why the perpendicular bisector through any vertex also contains the median, the altitude, and the angle bisector through that vertex. c. Explain why the incenter, circumcenter, centroid, and orthocenter are the same point. 45. Use coordinates to show that the lines containing ̶̶ RS, ̶̶ ST, and the altitudes of a triangle are concurrent. a. Find the slopes of b. Find the slopes of lines ℓ, m, and n. c. Write equations for lines ℓ, m, and n. d. Solve a system of equations to find the point P where lines ℓ and m intersect. ̶̶ RT. e. Show that line n contains P. f. What conclusion can you draw? 5- 3 Medians and Altitudes of Triangles 319 319 �� SPIRAL REVIEW 46. At a baseball game, a bag of peanuts costs $0.75 more than a bag of popcorn. If a family purchases 5 bags of peanuts and 3 bags of popcorn for $21.75, how much does one bag of peanuts cost? (Previous course) Determine if each biconditional is true. If false, give a counterexample. (Less
on 2-4) 47. The area of a rectangle is 40 cm 2 if and only if the length of the rectangle is 4 cm and the width of the rectangle is 10 cm. 48. A nonzero number n is positive if and only if -n is negative. ̶̶ QP, and ̶̶ NQ, Find each measure. (Lesson 5-2) ̶̶̶ QM are perpendicular bisectors of △JKL. 49. KL 50. QJ 51. m∠JQL Construction Orthocenter of a Triangle    Draw a large scalene acute triangle ABC on a piece of patty paper. Find the altitude of each side by folding the side so that it overlaps itself and so that the fold intersects the opposite vertex. Mark the point where the three lines containing the altitudes intersect and label it P. P is the orthocenter of △ABC. 1. Repeat the construction for a scalene obtuse triangle 2. Make a conjecture about the location of the and a scalene right triangle. orthocenter in an acute, an obtuse, and a right triangle. Q: What high school math classes did you take? A: Algebra 1, Geometry, and Statistics. KEYWORD: MG7 Career Q: What type of training did you receive? A: In high school, I took classes in electricity, electronics, and drafting. I began an apprenticeship program last year to prepare for the exam to get my license. Q: How do you use math? A: Determining the locations of outlets and circuits on blueprints requires good spatial sense. I also use ratios and proportions, calculate distances, work with formulas, and estimate job costs. Alex Peralta Electrician 320 320 Chapter 5 Properties and Attributes of Triangles �� 5-3 Special Points in Triangles In this lab you will use geometry software to explore properties of the four points of concurrency you have studied. Use with Lesson 5-3 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.B KEYWORD: MG7 Lab5 Activity 1 Construct a triangle. 2 Construct the perpendicular bisector of each side of the triangle. Construct the point of intersection of these three lines. This is the circumcenter of the triangle. Label it U and hide the
perpendicular bisectors. 3 4 5 In the same triangle, construct the bisector of each angle. Construct the point of intersection of these three lines. This is the incenter of the triangle. Label it I and hide the angle bisectors. In the same triangle, construct the midpoint of each side. Then construct the three medians. Construct the point of intersection of these three lines. Label the centroid C and hide the medians. In the same triangle, construct the altitude to each side. Construct the point of intersection of these three lines. Label the orthocenter O and hide the altitudes. 6 Move a vertex of the triangle and observe the positions of the four points of concurrency. In 1765, Swiss mathematician Leonhard Euler showed that three of these points are always collinear. The line containing them is called the Euler line. Try This 1. Which three points of concurrency lie on the Euler line? 2. Make a Conjecture Which point on the Euler line is always between the other two? Measure the distances between the points. Make a conjecture about the relationship of the distances between these three points. 3. Make a Conjecture Move a vertex of the triangle until all four points of concurrency are collinear. In what type of triangle are all four points of concurrency on the Euler line? 4. Make a Conjecture Find a triangle in which all four points of concurrency coincide. What type of triangle has this special property? 5- 3 Technology Lab 321 321 5-4 The Triangle Midsegment Theorem TEKS G.7.B Dimensionality and the geometry of location: use slopes and equations of lines to investigate... special segments of triangles and other polygons. Objective Prove and use properties of triangle midsegments. Vocabulary midsegment of a triangle Why learn this? You can use triangle midsegments to make indirect measurements of distances, such as the distance across a volcano. (See Example 3.) Also G.2.A, G.2.B, G.3.B, G.5.A, G.9.B A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. E X A M P L E 1 Examining Midsegments in the Coordinate Plane In △GHJ, show that midsegment parallel to GJ
and that KL = 1 __ ̶̶ Step 1 Find the coordinates of K and L. GJ. 2 ̶̶ KL is mdpt. of ̶̶̶ GH = ( ) -7 + (-5) -2 + 6 _ _, 2 2 = (-6, 2) mdpt. of ̶̶ HJ = ( 6 + 2 -5 + 1 _ _, 2 2 ) = (-2, 4) Step 2 Compare the slopes of slope of ̶̶ KL = ̶̶ GJ. ̶̶ KL and = 1 _ 2 4 - 2 _ -2 - (-6) slope of ̶̶ GJ = 2 - (-2) _ 1 - (-7) = 1 _ 2 Since the slopes are the same, ̶̶ KL and Step 3 Compare the lengths of ̶̶ KL ǁ ̶̶ GJ. ̶̶ GJ. 2 + (4 - 2) 2 = 2 √  5 KL = √  ⎤ ⎦ ⎡ ⎣ -2 - (-6) GJ = √  ⎤ ⎦ ⎤ ⎦ ⎡ ⎣ ⎡ ⎣ 2 - (-2) 1 - (-7) + (4 √  5 ), KL = 1 _ Since 2 √  5 = 1 _ 2 2 GJ. 2 2 = 4 √  5 1. The vertices of △RST are R (-7, 0), S (-3, 6), and T (9, 2). ̶̶ RT, and N is the midpoint of ̶̶ ST. M is the midpoint of Show that ̶̶̶ MN ǁ ̶̶ RS and MN = 1 __ 2 RS. 322 322 Chapter 5 Properties and Attributes of Triangles �� The relationship shown in Example 1 is true for the three midsegments of every triangle. Theorem 5-4-1 Triangle Midsegment Theorem
A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. ̶̶ DE ǁ ̶̶ AC, DE = 1_ 2 AC You will prove Theorem 5-4-1 in Exercise 38. E X A M P L E 2 Using the Triangle Midsegment Theorem Find each measure. A UW ST UW = 1 _ 2 UW = 1 _ (7.4) 2 UW = 3.7 △ Midsegment Thm. Substitute 7.4 for ST. Simplify. B m∠SVU ̶̶ ̶̶̶ UW ǁ ST m∠SVU = m∠VUW m∠SVU = 41° △ Midsegment Thm. Alt. Int.  Thm. Substitute 41° for m∠VUW. Find each measure. 2a. JL 2b. PM 2c. m∠MLK E X A M P L E 3 Indirect Measurement Application Anna wants to find the distance across the base of Capulin Volcano, an extinct volcano in New Mexico. She measures a triangle at one side of the volcano as shown in the diagram. What is AE? BD = 1 _ 2 775 = 1 _ 2 AE AE △ Midsegment Thm. Substitute 775 for BD. 1550 = AE Multiply both sides by 2. The distance AE across the base of the volcano is about 1550 meters. 3. What if…? Suppose Anna’s result in Example 3 is correct. To check it, she measures a second triangle. How many meters will she measure between H and F? 323 5- 4 The Triangle Midsegment Theorem 323 640 m1005 m640 m1005 mge07se_c05l04007aGFHAE��700 m920 m920 m775 m700 mge07se_c05l04006aCDBAE THINK AND DISCUSS 1. Explain why ̶̶ XY is NOT a midsegment of the triangle. 2. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of a triangle midsegment and list its properties. Then draw an example and a nonexample. 5-4 Exercises Exercises KEYWORD: MG7 5-4 KEYWORD:
MG7 Parent GUIDED PRACTICE 1. Vocabulary The midsegment of a triangle joins the triangle. (endpoints or midpoints)? of two sides of the ̶̶̶̶. 322 2. The vertices of △PQR are P (-4, -1), Q (2, 9), and R (6, 3). S is the midpoint of and T is the midpoint of ̶̶ QR. Show that ̶̶ ST ǁ ̶̶ PR and ST = 1 __ 2 PR. ̶̶ PQ, Find each measure. p. 323 3. NM 5. NZ 7. m∠YXZ 4. XZ 6. m∠LMN 8. m∠XLM. 323 Independent Practice For See Exercises Example 10 11–16 17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 9. Architecture In this A-frame house, ̶̶ XZ is 30 feet. ̶̶ CD is slightly above the width of the first floor The second floor and parallel to the midsegment of △XYZ. Is the width of the second floor more or less than 5 yards? Explain. � � � � � PRACTICE AND PROBLEM SOLVING 10. The vertices of △ABC are A (-6, 11), B (6, -3), and C (-2, -5). D is the midpoint ̶̶ AB. Show that ̶̶ AC, and E is the midpoint of ̶̶ DE ǁ ̶̶ CB and DE = 1 __ 2 CB. of Find each measure. 11. GJ 13. RJ 12. RQ 14. m∠PQR 15. m∠HGJ 16. m∠GPQ 324 324 Chapter 5 Properties and Attributes of Triangles �� 17. Carpentry In each support for the garden swing, ̶̶ BA and ̶̶ DE is attached at the midpoints ̶̶ BC. The distance AC is 4 1 __ 2 feet. the crossbar of legs The carpenter has a timber that is 30 inches long. Is this timber long enough to be used as one of the crossbars? Explain. △K
LM is the midsegment triangle of △GHJ. 18. What is the perimeter of △GHJ? 19. What is the perimeter of △KLM? 20. What is the relationship between the perimeter of △GHJ and the perimeter of △KLM? Algebra Find the value of n in each triangle. 21. 24. 22. 25. 23. 26. 27. /////ERROR ANALYSIS///// Below are two solutions for finding BC. Which is incorrect? Explain the error. 28. Critical Thinking Draw scalene △DEF. Label X as the midpoint of Y as the midpoint of midpoints. List all of the congruent angles in your drawing. ̶̶ EF, and Z as the midpoint of ̶̶ DE, ̶̶ DF. Connect the three 29. Estimation The diagram shows the sketch for a new street. Parallel parking spaces will be painted on both sides of the street. Each parallel parking space is 23 feet long. About how many parking spaces can the city accommodate on both sides of the new street? Explain your answer. ̶̶ EH, and ̶̶ CG, and △GHE, respectively. Find each measure. ̶̶ FJ are midsegments of △ABD, △GCD, 30. CG 31. EH 32. FJ 33. m∠DCG 34. m∠GHE 35. m∠FJH 36. Write About It An isosceles triangle has two congruent sides. Does it also have two congruent midsegments? Explain. 325 5- 4 The Triangle Midsegment Theorem 325 ��ge07sec05l04004aABoehmMarket Street (440 ft)Springfield RoadLake AvenueNew street�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 328. The figure shows the roads connecting towns A, B, and C. A music company has a store in each town and a distribution warehouse W at the midpoint of road a. What is the distance from the warehouse to point X? b. A truck starts at the warehouse, delivers instruments to the stores in towns A, B, and C (
in this order) and then returns to the warehouse. What is the total length of the trip, assuming the driver takes the shortest possible route? ̶̶ XY. 38. Use coordinates to prove the Triangle Midsegment Theorem. a. M is the midpoint of b. N is the midpoint of ̶̶ c. Find the slopes of PR and ̶̶ PQ. What are its coordinates? ̶̶ QR. What are its coordinates? ̶̶̶ MN. What can you conclude? d. Find PR and MN. What can you conclude? 39. ̶̶ PQ is a midsegment of △RST. What is the length of ̶̶ RT? 9 meters 21 meters 45 meters 63 meters 40. In △UVW, M is the midpoint of ̶̶̶ VW. Which statement is true? midpoint of ̶̶ VU, and N is the VM = VN MN = UV VU = 2VM VW = 1 _ 2 VN 41. △XYZ is the midsegment triangle of △JKL, XY = 8, YK = 14, and m∠YKZ = 67°. Which of the following measures CANNOT be determined? KL JY m∠XZL m∠KZY CHALLENGE AND EXTEND 42. Multi-Step The midpoints of the sides of a triangle are A (-6, 3), B (2, 1), and C (0, -3). Find the coordinates of the vertices of the triangle. 43. Critical Thinking Classify the midsegment triangle of an equilateral triangle by its side lengths and angle measures. Algebra Find the value of n in each triangle. 44. 45. 326 326 Chapter 5 Properties and Attributes of Triangles �� 46. △XYZ is the midsegment triangle of △PQR. Write a congruence statement involving all four of the smaller triangles. What is the relationship between the area of △XYZ and △PQR? ̶̶ AB is a midsegment of △XYZ. is a midsegment of △CDZ, and a. Copy and complete the table. 47. ̶̶ CD is a midsegment of △ABZ
. ̶̶̶ GH is a midsegment of △EFZ. ̶̶ EF 1 2 3 4 Number of Midsegment Length of Midsegment b. If this pattern continues, what will be the length of midsegment 8? c. Write an algebraic expression to represent the length of midsegment n. (Hint: Think of the midsegment lengths as powers of 2.) SPIRAL REVIEW Suppose a 2% acid solution is mixed with a 3% acid solution. Find the percent of acid in each mixture. (Previous course) 48. a mixture that contains an equal amount of 2% acid solution and 3% acid solution 49. a mixture that contains 3 times more 2% acid solution than 3% acid solution A figure has vertices G (-3, -2), H (0, 0), J (4, 1), and K (1, -2). Given the coordinates of the image of G under a translation, find the coordinates of the images of H, J, and K. (Lesson 1-7) 50. (-3, 2) 51. (1, -4) 52. (3, 0) Find each length. (Lesson 5-3) 53. NX 54. MR 55. NP Construction Midsegment of a Triangle    Draw a large triangle. Label the vertices A, B, and C. ̶̶ AB ̶̶ BC. Label the midpoints X Construct the midpoints of and and Y, respectively. Draw the midsegment ̶̶ XY. 1. Using a ruler, measure ̶̶ XY and ̶̶ AC. How are the two lengths related? 2. How can you use a protractor to verify that ̶̶ XY is parallel to ̶̶ AC? 327 5- 4 The Triangle Midsegment Theorem 327 �� SECTION 5A Segments in Triangles Location Contemplation A chain of music stores has locations in Ashville, Benton, and Carson. The directors of the company are using a coordinate plane to decide on the location for a new distribution warehouse. Each unit on the plane represents one mile. 1. A plot of land is available at the centroid of the triangle formed by the three cities. What are the coordinates for this location? 2. If the directors build the warehouse at the centroid, about how far will
it be from each of the cities? 3. Another plot of land is available at the orthocenter of the triangle. What are the coordinates for this location? 4. About how far would the warehouse be from each city if it were built at the orthocenter? 5. A third option is to build the warehouse at the circumcenter of the triangle. What are the coordinates for this location? 6. About how far would the warehouse be from each city if it were built at the circumcenter? 7. The directors decide that the warehouse should be equidistant from each city. Which location should they choose? 328 328 Chapter 5 Properties and Attributes of Triangles �� SECTION 5A Quiz for Lessons 5-1 Through 5-4 5-1 Perpendicular and Angle Bisectors Find each measure. 1. PQ 2. JM 3. AC 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints M (-1, -3) and N (7, 1). 5-2 Bisectors of Triangles 5. ̶̶ PY, and ̶̶ PX, bisectors of △RST. Find PS and XT. ̶̶ PZ are the perpendicular 6. ̶̶ JK and ̶̶ HK are angle bisectors of △GHJ. Find m∠GJK and the distance from K to ̶̶ HJ. 7. Find the circumcenter of △TVO with vertices T (9, 0), V (0, -4), and O (0, 0). 5-3 Medians and Altitudes of Triangles 8. In △DEF, BD = 87, and WE = 38. Find BW, CW, and CE. 9. Paula cuts a triangle with vertices at coordinates (0, 4), (8, 0), and (10, 8) from grid paper. At what coordinates should she place the tip of a pencil to balance the triangle? 10. Find the orthocenter of △PSV with vertices P (2, 4), S (8, 4), and V (4, 0). 5-4 The Triangle Midsegment Theorem 11. Find ZV, PM, and m∠RZV in △JMP. 12. What is the distance XZ across
the pond? � �� Ready to Go On? 329 329 �� Solving Compound Inequalities Algebra To solve an inequality, you use the Properties of Inequality and inverse operations to undo the operations in the inequality one at a time. See Skills Bank page S60 Properties of Inequality PROPERTY ALGEBRA Addition Property Subtraction Property Multiplication Property Division Property Transitive Property Comparison Property If a < b, then a + c < b + c. If a < b, then a - c < b - c. If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. If a < b and c > 0, then a _ c < b _ c. If a < b and c < 0, then a _ c > b _ c. If a < b and b < c, then a < c. If a + b = c and b > 0, then a < c. A compound inequality is formed when two simple inequalities are combined into one statement with the word and or or. To solve a compound inequality, solve each simple inequality and find the intersection or union of the solutions. The graph of a compound inequality may represent a line, a ray, two rays, or a segment. Example Solve the compound inequality 5 < 20 - 3a ≤ 11. What geometric figure does the graph represent? 5 < 20 - 3a AND 20 - 3a ≤ 11 Rewrite the compound inequality as two -15 < -3a 5 > a AND AND -3a ≤ -9 Subtract 20 from both sides. simple inequalities. a ≥ 3 Divide both sides by -3 and reverse the inequality symbols. 3 ≤ a < 5 Combine the two solutions into a single statement. The graph represents a segment. Try This TAKS Grades 9–11 Obj. 4 Solve. What geometric figure does each graph represent? 1. -4 + x > 1 OR -8 + 2x < -6 2. 2x - 3 ≥ -5 OR x - 4 > -1 3. -6 < 7 - x ≤ 12 5. 3x ≥ 0 OR x + 5 < 7 4. 22 < -2 - 2x ≤ 54 6. 2x - 3 ≤ 5 OR -2x + 3 ≤
-9 330 330 Chapter 5 Properties and Attributes of Triangles �� 5-5 Use with Lesson 5-5 Activity 1 Explore Triangle Inequalities Many of the triangle relationships you have learned so far involve a statement of equality. For example, the circumcenter of a triangle is equidistant from the vertices of the triangle, and the incenter is equidistant from the sides of the triangle. Now you will investigate some triangle relationships that involve inequalities. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about... polygons and their component parts based on explorations and concrete models. Also G.5.B 1 Draw a large scalene triangle. Label the vertices A, B, and C. 2 Measure the sides and the angles. Copy the table below and record the measures in the first row. BC AC AB m∠A m∠B m∠C Triangle 1 Triangle 2 Triangle 3 Triangle 4 Try This 1. In the table, draw a circle around the longest side length, and draw a circle around the greatest angle measure of △ABC. Draw a square around the shortest side length, and draw a square around the least angle measure. 2. Make a Conjecture Where is the longest side in relation to the largest angle? Where is the shortest side in relation to the smallest angle? 3. Draw three more scalene triangles and record the measures in the table. Does your conjecture hold? Activity 2 1 Cut three sets of chenille stems to the following lengths. 3 inches, 4 inches, 6 inches 3 inches, 4 inches, 7 inches 3 inches, 4 inches, 8 inches 2 Try to make a triangle with each set of chenille stems. Try This 4. Which sets of chenille stems make a triangle? 5. Make a Conjecture For each set of chenille stems, compare the sum of any two lengths with the third length. What is the relationship? 6. Select a different set of three lengths and test your conjecture. Does your conjecture hold? 5- 5 Geometry Lab 331 331 5-5 Indirect Proof and Inequalities in One Triangle TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.C, G.3.E, G.5.B Objectives Write indirect proofs. Apply inequalities in one triangle. Why learn this? You can use a triangle inequality to find a reasonable range
of values for an unknown distance. (See Example 5.) Vocabulary indirect proof So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Writing an Indirect Proof 1. Identify the conjecture to be proven. 2. Assume the opposite (the negation) of the conclusion is true. 3. Use direct reasoning to show that the assumption leads to a contradiction. 4. Conclude that since the assumption is false, the original conjecture must be true. E X A M P L E 1 Writing an Indirect Proof Write an indirect proof that a right triangle cannot have an obtuse angle. Step 1 Identify the conjecture to be proven. Given: △JKL is a right triangle. Prove: △JKL does not have an obtuse angle. Step 2 Assume the opposite of the conclusion. Assume △JKL has an obtuse angle. Let ∠K be obtuse. Step 3 Use direct reasoning to lead to a contradiction. m∠K + m∠L = 90° The acute  of a rt. △ are comp. m∠K = 90° - m∠L m∠K > 90° Subtr. Prop. of = Def. of obtuse ∠ 90° - m∠L > 90° m∠L < 0° Substitute 90° - m∠L for m∠K. Subtract 90° from both sides and solve for m∠L. However, by the Protractor Postulate, a triangle cannot have an angle with a measure less than 0°. Step 4 Conclude that the original conjecture is true. The assumption that △JKL has an obtuse angle is false. Therefore △JKL does not have an obtuse angle. 1. Write an indirect proof that a triangle cannot have two right angles. 332 332 Chapter 5 Properties and Attributes of Triangles �� The positions of the longest and shortest sides of a triangle are related to the positions of the largest
and smallest angles. Theorems Angle-Side Relationships in Triangles THEOREM HYPOTHESIS CONCLUSION 5-5-1 If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. (In △, larger ∠ is opp. longer side.) 5-5-2 If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (In △, longer side is opp. larger ∠.) m∠C > m∠A XY > XZ AB > BC m∠Z > m∠Y You will prove Theorem 5-5-1 in Exercise 67. PROOF PROOF Theorem 5-5-2 Given: m∠P > m∠R Prove: QR > QP Consider all cases when you assume the opposite. If the conclusion is QR > QP, the negation includes QR < QP and QR = QP. Indirect Proof: Assume QR ≯ QP. This means that either QR < QP or QR = QP. Case 1 the longer side. This contradicts the given information. So QR ≮ QP. If QR < QP, then m∠P < m∠R because the larger angle is opposite Case 2 This also contradicts the given information, so QR ≠ QP. If QR = QP, then m∠P = m∠R by the Isosceles Triangle Theorem. The assumption QR ≯ QP is false. Therefore QR > QP. E X A M P L E 2 Ordering Triangle Side Lengths and Angle Measures A Write the angles in order from smallest to largest. ̶̶ GJ, so the smallest angle is ∠H. ̶̶ HJ, so the largest angle is ∠G. The shortest side is The longest side is The angles from smallest to largest are ∠H, ∠J, and ∠G. B Write the sides in order from shortest to longest. m∠M = 180° - (39° + 54°) = 87° △ Sum Thm. ̶̶̶ The smallest angle is ∠L, so the shortest side is KM. ̶̶ KL. The largest angle is ∠M, so the longest side is ̶̶̶ LM, and The sides from shortest to longest are ̶̶̶
KM, ̶̶ KL. 2a. Write the angles in order from smallest to largest. 2b. Write the sides in order from shortest to longest. 5- 5 Indirect Proof and Inequalities in One Triangle 333 333 �� A triangle is formed by three segments, but not every set of three segments can form a triangle. Segments with lengths of 7, 4, and 4 can form a triangle. Segments with lengths of 7, 3, and 3 cannot form a triangle. A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Theorem 5-5-3 Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. AB + BC > AC BC + AC > AB AC + AB > BC You will prove Theorem 5-5-3 in Exercise 68. E X A M P L E 3 Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. A 3, 5 ✓ 10 > 5 ✓ 12 > 3 ✓ Yes—the sum of each pair of lengths is greater than the third length. B 4, 6.5, 11 4 + 6.5  11 10.5 ≯ 11 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. C n + 5, n 2, 2n, when n = 3 Step 1 Evaluate each expression when n = 3 2n 2 (3) 6 To show that three lengths cannot be the side lengths of a triangle, you only need to show that one of the three triangle inequalities is false. Step 2 Compare the lengths 17 > 6 ✓ 14 > 9 ✓ 15 > 8 ✓ Yes—the sum of each pair of lengths is greater than the third length. Tell whether a triangle can have sides with the given lengths. Explain. 3a. 8, 13, 21 3c. t - 2, 4t, t 2 + 1, when t = 4 3b. 6.2, 7, 9 334 334 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 4 Finding Side Lengths The lengths of two sides of a triangle are 6 centimeters and 11 centimeters. Find the range of possible lengths for the third side. Let s represent the length of the third side. Then apply the Triangle Inequality Theorem. s + 6 > 11 s
+ 11 > 6 s > 5 s > -5 6 + 11 > s 17 > s Combine the inequalities. So 5 < s < 17. The length of the third side is greater than 5 centimeters and less than 17 centimeters. 4. The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side. E X A M P L E 5 Travel Application The map shows the approximate distances from San Antonio to Mason and from San Antonio to Austin. What is the range of distances from Mason to Austin? Let d be the distance from Mason to Austin. d + 111 > 78 d + 78 > 111 111 + 78 > d △ Inequal. Thm. d > -33 d > 33 189 > d Subtr. Prop. of Inequal. 33 < d < 189 Combine the inequalities. The distance from Mason to Austin is greater than 33 miles and less than 189 miles. 5. The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City? THINK AND DISCUSS 1. To write an indirect proof that an angle is obtuse, a student assumes that the angle is acute. Is this the correct assumption? Explain. 2. Give an example of three measures that can be the lengths of the sides of a triangle. Give an example of three lengths that cannot be the sides of a triangle. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, explain what you know about △ABC as a result of the theorem. 5- 5 Indirect Proof and Inequalities in One Triangle 335 335 AustinSanMarcosJohnsonCitySeguinSan AntonioMason10103535290281377183879078 mi111 mige07se_c05l05002aABeckmann�� 5-5 Exercises Exercises KEYWORD: MG7 5-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe the process of an indirect proof in your own words Write an indirect proof of each statement. p. 332 2. A scalene triangle cannot have two congruent angles. 3. An isosceles triangle cannot have a base angle that is a right angle. Write the angles in order p. 333 from smallest
to largest. 5. Write the sides in order from shortest to longest Tell whether a triangle can have sides with the given lengths. Explain. p. 334 6. 4, 7, 10 7. 2, 9, 12, 3 1 _ 8. 3 1 _ 2 2, 6 9. 3, 1.1, 1.7 10. 3x, 2x - 1, x 2, when x = 5 11. 7c + 6, 10c - 7, 3 c 2, when. 335 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. 12. 8 mm, 12 mm 13. 16 ft, 16 ft 14. 11.4 cm, 12 cm. 335 15. Design The refrigerator, stove, and sink in a kitchen are at the vertices of a path called the work triangle. a. If the angle at the sink is the largest, which side of the work triangle will be the longest? b. The designer wants the longest side of this triangle to be 9 feet long. Can the lengths of the other sides be 5 feet and 4 feet? Explain. �� Independent Practice Write an indirect proof of each statement. PRACTICE AND PROBLEM SOLVING 16. A scalene triangle cannot have two congruent midsegments. 17. Two supplementary angles cannot both be obtuse angles. 18. Write the angles in order from smallest to largest. �� 19. Write the sides in order �� from shortest to longest. For See Exercises Example 16–17 18–19 20–25 26–31 32 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 Tell whether a triangle can have sides with the given lengths. Explain. 20. 6, 10, 15 21. 14, 18, 32 22. 11.9, 5.8, 5.8 23. 103, 41.9, 62.5 24. z + 8, 3z + 5, 4z - 11, when z = 6 25. m + 11, 8m, m 2 + 1, when m = 3 336 336 Chapter 5 Properties and Attributes of Triangles �� The lengths of two sides of a triangle are given. Find the range of possible lengths The lengths of two sides of a triangle are given. Find the range of possible lengths for the
third side. for the third side. Bicycles 26. 26. 4 yd, 19 yd 29. 3.07 m, 1.89 m 27. 28 km, 23 km in., 3 5_ 30. 2 1_ 8 8 in. 28. 9.2 cm, 3.8 cm 31. 3 5_ ft, 6 1_ 2 6 ft 32. Bicycles The five steel tubes of this mountain bike frame form two triangles. List the five tubes in order from shortest to longest. Explain your answer. 33. Critical Thinking The length of the base of an isosceles triangle is 15. What is the range of possible lengths for each leg? Explain. List the sides of each triangle in order from shortest to longest. 34. 35. Lance Armstrong, of Austin, broke his own record in 2005 and became the first person to win seven consecutive titles in the Tour de France cycling competition. Only one other person has won five consecutive times. In each set of statements, name the two that contradict each other. 36. △PQR is a right triangle. 37. ∠Y is supplementary to ∠Z. △PQR is a scalene triangle. △PQR is an acute triangle. 38. △JKL is isosceles with base ̶̶ JL. In △JKL, m∠K > m∠J In △JKL, JK > LK 40. Figure A is a polygon. Figure A is a triangle. Figure A is a quadrilateral. m∠Y < 90° ∠Y is an obtuse angle. 39. ̶̶ AB ⊥ ̶̶ AB ≅ ̶̶ AB ǁ ̶̶ BC ̶̶ CD ̶̶ BC 41. x is even. x is a multiple of 4. x is prime. Compare. Write <, >, or =. PS 42. QS 44. QS 46. PQ QR RS 43. PQ QS 45. QS 47. RS RS PS 48. m∠ABE m∠BEA 49. m∠CBE m∠CEB 50. m∠DCE m∠DEC 51. m∠DCE m∠CDE 52. m∠ABE m∠EAB 53. m∠EBC m∠ECB List the
angles of △JKL in order from smallest to largest. 54. J (-3, -2), K (3, 6), L (8, -2) 55. J (-5, -10), K (-5, 2), L (7, -5) 56. J (-4, 1), K (-3, 8), L (3, 4) 57. J (-10, -4), K (0, 3), L (2, -8) 58. Critical Thinking An attorney argues that her client did not commit a burglary because a witness saw her client in a different city at the time of the burglary. Explain how this situation is an example of indirect reasoning. 5- 5 Indirect Proof and Inequalities in One Triangle 337 337 B54.1 cm50.8 cmADCge07se_ c05105005aa1st Pass2/7/05N Patel64º66º56º50º�� 59. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes between four cities. a. The airline’s planes fly at an average speed of 500 mi/h. What is the range of time it might take to fly from Auburn (A) to Raymond (R)? b. The airline offers one frequent-flier mile for every mile flown. Is it possible to earn 1800 miles by flying from Millford (M) to Auburn (A)? Explain. Multi-Step Each set of expressions represents the lengths of the sides of a triangle. Find the range of possible values of n. 60. n, 6, 8 61. 2n, 5, 7 62. n + 1, 3, 6 63. n + 1, n + 2, n + 3 64. n + 2, n + 3, 3n - 2 65. n, n + 2, 2n + 1 66. Given that P is in the interior of △XYZ, prove that XY + XP + PZ > YZ. 67. Complete the proof of Theorem 5-5-1 by filling in the blanks. Given: RS > RQ Prove: m∠RQS > m∠S Proof: ̶̶ RS so that RP = RQ. So?, and m∠
1 = m∠2 by c. ̶̶̶̶ Locate P on by b. m∠RQS = d. Inequality. Then m∠RQS > m∠2 by e. m∠2 = m∠3 + f. Inequality. Therefore m∠RQS > m∠S by g.?. So m∠RQS > m∠1 by the Comparison Property of ̶̶̶̶?. So m∠2 > m∠S by the Comparison Property of ̶̶̶̶?. By the Exterior Angle Theorem, ̶̶̶̶ ̶̶ RQ by a. ̶̶ RP ≅?. By the Angle Addition Postulate, ̶̶̶̶?. Then ∠1 ≅ ∠2 ̶̶̶̶?. ̶̶̶̶ 68. Complete the proof of the Triangle Inequality Theorem. Given: △ABC Prove: AB + BC > AC, AB + AC > BC, AC + BC > AB Proof: One side of △ABC is as long as or longer than each of the other sides. Let this side be remains to be proved is AC + BC > AB. ̶̶ AB. Then AB + BC > AC, and AB + AC > BC. Therefore what Statements Reasons 1. a.? ̶̶̶̶ 2. Locate D on  AC so that BC = DC. 3. AC + DC = b.? ̶̶̶̶ 4. ∠1 ≅ ∠2 5. m∠1 = m∠2 6. m∠ABD = m∠2 + e.? ̶̶̶̶ 7. m∠ABD > m∠2 8. m∠ABD > m∠1 9. AD > AB 10. AC + DC > AB 11. i.? ̶̶̶̶ 1. Given 2. Ruler Post. 3. Seg. Add. Post. 4. c. 5. d.? ̶̶̶̶? ̶̶̶̶ 6. ∠ Add. Post. 7. Comparison Prop. of Inequal. 8. f. 9. g. 10. h.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ 11. Subst. 69.
Write About It Explain why the hypotenuse is always the longest side of a right triangle. Explain why the diagonal of a square is longer than each side. 338 338 Chapter 5 Properties and Attributes of Triangles �� 70. The lengths of two sides of a triangle are 3 feet and 5 feet. Which could be the length of the third side? 3 feet 8 feet 15 feet 16 feet 71. Which statement about △GHJ is false? GH < GJ m∠H > m∠ J GH + HJ < GJ △GHJ is a scalene triangle. 72. In △RST, m∠S = 92°. Which is the longest side of △RST? ̶̶ RS ̶̶ ST ̶̶ RT Cannot be determined CHALLENGE AND EXTEND 73. Probability A bag contains five sticks. The lengths of the sticks are 1 inch, 3 inches, 5 inches, 7 inches, and 9 inches. Suppose you pick three sticks from the bag at random. What is the probability you can form a triangle with the three sticks? 74. Complete this indirect argument that √  2 is irrational. Assume that a.?. ̶̶̶̶ __ q, where p and q are positive integers that have no common factors.?, and p 2 = c. ̶̶̶̶ p Then √  2 = Thus 2 = b. p is even. Since p 2 is the square of an even number, p 2 is divisible by 4 because d.?, and so q is even. Then p and ̶̶̶̶ q have a common factor of 2, which contradicts the assumption that p and q have no common factors.?. But then q 2 must be even because e. ̶̶̶̶?. This implies that p 2 is even, and thus ̶̶̶̶ 75. Prove that the perpendicular segment from a point to a line is the shortest segment from the point to the line. ̶̶ PX ⊥ ℓ. Y is any point on ℓ other than X. Given: Prove: PY > PX Plan: Show that ∠2 and ∠P are complementary. Use the Comparison Property of Inequality to show that 90° > m∠2. Then show that m∠1 > m∠2 and thus PY > PX.
SPIRAL REVIEW Write the equation of each line in standard form. (Previous course) 76. the line through points (-3, 2) and (-1, -2) 77. the line with slope 2 and x-intercept of -3 Show that the triangles are congruent for the given value of the variable. (Lesson 4-4) 78. △PQR ≅ △TUS, when x = -1 79. △ABC ≅ △EFD, when p = 6 Find the orthocenter of a triangle with the given vertices. (Lesson 5-3) 80. R (0, 5), S (4, 3), T (0, 1) 81. M (0, 0), N (3, 0), P (0, 5) 5- 5 Indirect Proof and Inequalities in One Triangle 339 339 �� 5-6 Inequalities in Two Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.E Objective Apply inequalities in two triangles. Who uses this? Designers of this circular swing ride can use the angle of the swings to determine how high the chairs will be at full speed. (See Example 2.) In this lesson, you will apply inequality relationships between two triangles. Theorems Inequalities in Two Triangles THEOREM HYPOTHESIS CONCLUSION 5-6-1 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. 5-6-2 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. m∠A > m∠D BC > EF m∠ J > m∠M GH > KL You will prove Theorem 5-6-1 in Exercise 35. PROOF PROOF Converse of the Hinge Theorem ̶̶ PR ≅ ̶̶ XZ, QR > YZ ̶̶ PQ ≅ Given: Prove: m∠P > m
∠X ̶̶ XY, Indirect Proof: Assume m∠P ≯ m∠X. So either m∠P < m∠X, or m∠P = m∠X. Case 1 If m∠P < m∠X, then QR < YZ by the Hinge Theorem. This contradicts the given information that QR > YZ. So m∠P ≮ m∠X. Case 2 If m∠P = m∠X, then ∠P ≅ ∠X. So △PQR ≅ △XYZ by SAS. Then ̶̶ YZ by CPCTC, and QR = YZ. This also contradicts ̶̶ QR ≅ the given information. So m∠P ≠ m∠X. The assumption m∠P ≯ m∠X is false. Therefore m∠P > m∠X. 340 340 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Hinge Theorem and Its Converse A Compare m∠PQS and m∠RQS. Compare the side lengths in △PQS and △RQS. QS = QS PQ = RQ PS > RS By the Converse of the Hinge Theorem, m∠PQS > m∠RQS. B Compare KL and MN. Compare the sides and angles in △KLN and △MNL. KN = ML LN = LN m∠LNK < m∠NLM By the Hinge Theorem, KL < MN. C Find the range of values for z. Step 1 Compare the side lengths in △TUV and △TWV. TV = TV VU = VW TU < TW By the Converse of the Hinge Theorem, m∠UVT < m∠WVT. 6z - 3 < 45 Substitute the given values. z < 8 Add 3 to both sides and divide both sides by 6. Step 2 Since ∠UVT is in a triangle, m∠UVT > 0°. 6z - 3 > 0 Substitute the given value. z > 0.5 Add 3 to both sides and divide both sides by 6. Step 3 Combine the inequalities. The range of values for z is 0.5
< z < 8. Compare the given measures. 1a. m∠EGH and m∠EGF 1b. BC and AB E X A M P L E 2 Entertainment Application The angle of the swings in a circular swing ride changes with the speed of the ride. The diagram shows the position of one swing at two different speeds. Which rider is farther from the base of the swing tower? Explain. The height of the tower and the length of the cable holding the chair are the same in both triangles. The angle formed by the swing in position A is smaller than the angle formed by the swing in position B. So rider B is farther from the base of the tower than rider A by the Hinge Theorem. 5- 6 Inequalities in Two Triangles 341 341 ��BAge07sec05L06002_A 2. When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain. E X A M P L E 3 Proving Triangle Relationships Write a two-column proof. ̶̶ NL Given: Prove: KM > NM ̶̶ KL ≅ Proof: ̶̶ KL ≅ ̶̶̶ LM ≅ ̶̶ NL ̶̶̶ LM 1. 2. Statements Reasons 1. Given 2. Reflex. Prop. of ≅ 3. m∠KLM = m∠NLM + m∠KLN 3. ∠ Add. Post. 4. m∠KLM > m∠NLM 4. Comparison Prop. of Inequal. 5. KM > NM 5. Hinge Thm. Write a two-column proof. 3a. Given: C is the midpoint of ̶̶ BD. m∠1 = m∠2 m∠3 > m∠4 Prove: AB > ED 3b. Given: ∠SRT ≅ ∠STR TU > RU Prove: m∠TSU > m∠RSU THINK AND DISCUSS 1. Describe a real-world object that shows the Hinge Theorem or its converse. 2. Can you make a conclusion about the triangles shown at right by applying the Hinge Theorem? Explain. 3. GET ORGANIZED Copy and complete
the graphic organizer. In each box, use the given triangles to write a statement for the theorem. 342 342 Chapter 5 Properties and Attributes of Triangles �� 5-6 Exercises Exercises KEYWORD: MG7 5-6 KEYWORD: MG7 Parent GUIDED PRACTICE Compare the given measures. p. 341 1. AC and XZ 2. m∠SRT and m∠QRT 3. KL and KN Find the range of values for x. 4. 5. 6. Health A therapist can take measurements to p. 341 gauge the flexibility of a patient’s elbow joint. In which position is the angle measure at the elbow joint greater? Explain. 342 Independent Practice For See Exercises Example 9–14 15 16 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 8. Write a two-column proof. Given: ̶̶ FH is a median of △DFG. m∠DHF > m∠GHF Prove: DF > GF PRACTICE AND PROBLEM SOLVING Compare the given measures. 9. m∠DCA and m∠BCA 10. m∠GHJ and m∠KLM 11. TU and SV Find the range of values for z. 12. 13. 14. 5- 6 Inequalities in Two Triangles 343 343 �� 15. Industry The operator of a backhoe changes the distance between the cab and the bucket by changing the angle formed by the arms. In which position is the distance from the cab to the bucket greater? Explain. �� 16. Write a two-column proof. ̶̶̶ MQ, JQ > NP ̶̶ KP ≅ ̶̶̶ NM, ̶̶ JK ≅ Given: Prove: m∠K > m∠M 17. Critical Thinking ABC is an isosceles triangle with base triangle with base ̶̶ YZ. Given that ̶̶ AB ≅ ̶̶ BC. XYZ is an is
osceles ̶̶ XY and m∠A = m∠X, compare BC and YZ. Compare. Write <, >, or =. 18. m∠QRP m∠SRP 19. m∠QPR m∠QRP 20. m∠PRS m∠RSP 21. m∠RSP m∠RPS 22. m∠QPR m∠RPS 23. m∠PSR m∠PQR Make a conclusion based on the Hinge Theorem or its converse. (Hint : Draw a sketch.) ̶̶ DE, ̶̶ RT. The endpoints of 25. △RST is isosceles with base 24. In △ABC and △DEF, ̶̶ BC ≅ ̶̶ AB ≅ ̶̶ EF, m∠B = 59°, and m∠E = 47°. ̶̶ SV are vertex S and a point V on ̶̶ RT. RV = 4, and TV = 5. 26. In △GHJ and △KLM, ̶̶̶ GH ≅ ̶̶ KL, and ̶̶ GJ ≅ ̶̶̶ KM. ∠G is a right angle, and ∠K is an acute angle. 27. In △XYZ, ̶̶̶ XM is the median to ̶̶ YZ, and YX > ZX. 28. Write About It The picture shows a door hinge in two different positions. Use the picture to explain why Theorem 5-6-1 is called the Hinge Theorem. 29. Write About It Compare the Hinge Theorem to the SAS Congruence Postulate. How are they alike? How are they different? 30. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The solid lines in the figure show an airline’s routes between four cities. a. A traveler wants to fly from Jackson (J) to Shelby (S), but there is no direct flight between these cities. Given that m∠NSJ < m∠HSJ, should the traveler first fly to Newton Springs (N) or to Hollis (H) if he wants to minimize the number of miles flown? Why? b. The distance from Shelby (S) to Jackson (J) is 182 mi. What is
the minimum number of miles the traveler will have to fly? 344 344 Chapter 5 Properties and Attributes of Triangles �� 31. ̶̶ ML is a median of △JKL. Which inequality best describes the range of values for x? x > 2 x > 10 < 10 32. ̶̶ DC is a median of △ABC. Which of the following statements is true? BC < AC BC > AC AD = DB DC = AB 33. Short Response Two groups start hiking from the same camp. Group A hikes 6.5 miles due west and then hikes 4 miles in the direction N 35° W. Group B hikes 6.5 miles due east and then hikes 4 miles in the direction N 45° E. At this point, which group is closer to the camp? Explain. CHALLENGE AND EXTEND 34. Multi-Step In △XYZ, XZ = 5x + 15, XY = 8x - 6, and m∠XVZ > m∠XVY. Find the range of values for x. 35. Use these steps to write a paragraph proof of the Hinge Theorem. ̶̶ BC ≅ ̶̶ AB ≅ ̶̶ DE, Given: ̶̶ EF, m∠ABC > m∠DEF Prove: AC > DF a. Locate P outside △ABC so that ∠ABP ≅ ∠DEF and thus ̶̶ BP ≅ ̶̶ AP ≅ ̶̶ EF. Show that △ABP ≅ △DEF and ̶̶ DF. ̶̶ AC so that Show that △BQP ≅ △BQC and thus ̶̶ BQ bisects ∠PBC. Draw ̶̶ QP ≅ ̶̶ QC. b. Locate Q on ̶̶ QP. c. Justify the statements AQ + QP > AP, AQ + QC = AC, AQ + QC > AP, AC > AP, and AC > DF. SPIRAL REVIEW Find the range and mode, if any, of each set of data. (Previous course) 36. 2, 5, 1, 0.5, 0.75, 2 37. 95, 97, 89, 87, 85, 99 38. 5, 5, 7, 9, 4, 4, 8, 7 For the given information, show that m
ǁ n. State any postulates or theorems used. (Lesson 3-3) 39. m∠2 = (3x + 21) °, m∠6 = (7x + 1) °, x = 5 40. m∠4 = (2x + 34) °, m∠7 = (15x + 27) °, x = 7 Find each measure. (Lesson 5-4) 41. DF 42. BC 43. m∠BFD 5- 6 Inequalities in Two Triangles 345 345 6.5 mi6.5 mi4 mi4 miABThird sketchge07se_c05106009aatopo mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� Simplest Radical Form Algebra When a problem involves square roots, you may be asked to give the answer in simplest radical form. Recall that the radicand is the expression under the radical sign. See Skills Bank page S55 Simplest Form of a Square-Root Expression An expression containing square roots is in simplest form when • the radicand has no perfect square factors other than 1. • the radicand has no fractions. • there are no square roots in any denominator. To simplify a radical expression, remember that the square root of a product is equal to the product of the square roots. Also, the square root of a quotient is equal to the quotient of the square roots. √  ab = √  a · √  b, when a ≥ 0 and, when a ≥ 0 and b > 0 Examples Write each expression in simplest radical form. A √  216 √  216 216 has a perfect-square factor of 36, so the expression is not in simplest radical form. √  (36) (6) Factor the radicand. √  36 · √  6 Product Property of Square Roots 6 √  6 Simplify. Try This TAKS Grades 9–11 Obj. 2 Write each expression in simplest radical form. 3. 10_ √  2 2. √3_ 1
. √720 16 346 346 Chapter 5 Properties and Attributes of Triangles ) There is a square root in the denominator, so the expression is not in simplest radical form. Multiply by a form of 1 to eliminate the square root in the denominator Simplify. Divide. 4. √1_ 3 5. √45 5-7 Hands-on Proof of the Pythagorean Theorem In Lesson 1-6, you used the Pythagorean Theorem to find the distance between two points in the coordinate plane. In this activity, you will build figures and compare their areas to justify the Pythagorean Theorem. Use with Lesson 5-7 TEKS G.8.C Congruence and the geometry of size: derive, extend, and use the Pythagorean Theorem. Also G.9.B Activity 1 Draw a large scalene right triangle on graph paper. Draw three copies of the triangle. On each triangle, label the shorter leg a, the longer leg b, and the hypotenuse c. 2 Draw a square with a side length of b - a. Label each side of the square. 3 Cut out the five figures. Arrange them to make the composite figure shown at right. 4 You can think of this composite figure as being made of the two squares outlined in red. What are the side length and area of the small red square? of the large red square? 5 Use your results from Step 4 to write an algebraic expression for the area of the composite figure. 6 Now rearrange the five figures to make a single square with side length c. Write an algebraic expression for the area of this square. Try This 1. Since the composite figure and the square with side length c are made of the same five shapes, their areas are equal. Write and simplify an equation to represent this relationship. What conclusion can you make? 2. Draw a scalene right triangle with different side lengths. Repeat the activity. Do you reach the same conclusion? 5- 7 Geometry Lab 347 347 5-7 The Pythagorean Theorem TEKS G.8.C Congruence and the geometry of size: derive, extend, and use the Pythagorean Theorem. Also G.1.B, G.5.B, G.5.D, G.11.C Objectives Use the Pythagorean Theorem and its converse to solve problems.
Use Pythagorean inequalities to classify triangles. Vocabulary Pythagorean triple Why learn this? You can use the Pythagorean Theorem to determine whether a ladder is in a safe position. (See Example 2.) The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. For more on the Pythagorean Theorem, see the Theorem Builder on page xxvi. � � � The Pythagorean Theorem is named for the Greek mathematician Pythagoras, who lived in the sixth century B.C.E. However, this relationship was known to earlier people, such as the Babylonians, Egyptians, and Chinese. There are many different proofs of the Pythagorean Theorem. The one below uses area and algebra. PROOF PROOF Pythagorean Theorem Given: A right triangle with leg lengths a and b and hypotenuse of length c Prove: a 2 + b 2 = c 2 The area A of a square with side length s is given by the formula A = s 2. The area A of a triangle with base b and height h is given by the formula A = 1 __ bh. 2 Proof: Arrange four copies of the triangle as shown. The sides of the triangles form two squares. The area of the outer square is (a + b) 2. The area of the inner square is c 2. The area of each blue triangle is 1 __ 2 ab. area of outer square = area of 4 blue triangles + area of inner square (a + b) 2 = 4 ( 1 _ 2 ab) + c 2 a 2 + 2ab + b 2 = 2ab + Substitute the areas. Simplify. Subtract 2ab from both sides. The Pythagorean Theorem gives you a way to find unknown side lengths when you know a triangle is a right triangle. 348 348 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form 52 = x 2 √  52 = x Pythagorean Theorem Substitute 6 for a, 4 for b, and x for c. Simplify. Find the positive square root. x = √ �
�� (4) (13) = 2 √  13 Simplify the radicalx - 1) 2 = x 2 Pythagorean Theorem Substitute 5 for a, x - 1 for b, 25 + x 2 - 2x + 1 = x 2 Multiply. and x for c. -2x + 26 = 0 Combine like terms. 26 = 2x x = 13 Add 2x to both sides. Divide both sides by 2. Find the value of x. Give your answer in simplest radical form. 1a. 1b. E X A M P L E 2 Safety Application To prevent a ladder from shifting, safety experts recommend that the ratio of a : b be 4 : 1. How far from the base of the wall should you place the foot of a 10-foot ladder? Round to the nearest inch. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4x is the distance in feet from the top of the ladder to the base of the wall4x) 2 + x 2 = 10 2 17 x 2 = 100 x 2 = 100 _ 17  100 _ 17 x = √ ≈ 2 ft 5 in. Pythagorean Theorem Substitute. Multiply and combine like terms. Divide both sides by 17. Find the positive square root and round it. 2. What if...? According to the recommended ratio, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch. A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple. Common Pythagorean Triples 3, 4, 5 5, 12, 13, 8, 15, 17 7, 24, 25 349 5- 7 The Pythagorean Theorem 349 ��abge07sec05l07002aAB E X A M P L E 3 Identifying Pythagorean Triples Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain 12 2 + b 2 = 15 2 b 2 = 81 b = 9 Pythagorean Theorem Substitute 12 for a and 15 for c. Multiply and subtract 144 from both sides. Find the positive square root. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2,
so they form a Pythagorean triple + 15 2 = c 2 306 = c 2 Pythagorean Theorem Substitute 9 for a and 15 for b. Multiply and add. c = √  306 = 3 √  34 Find the positive square root and simplify. The side lengths do not form a Pythagorean triple because 3 √  34 is not a whole number. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 3a. 3c. 3b. 3d. The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. Theorems 5-7-1 Converse of the Pythagorean Theorem THEOREM HYPOTHESIS CONCLUSION If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. △ABC is a right triangle. a 2 + b 2 = c 2 You will prove Theorem 5-7-1 in Exercise 45. 350 350 Chapter 5 Properties and Attributes of Triangles �� You can also use side lengths to classify a triangle as acute or obtuse. Theorems 5-7-2 Pythagorean Inequalities Theorem In △ABC, c is the length of the longest side. If c 2 > a 2 + b 2, then △ABC is an obtuse triangle. If c 2 < a 2 + b 2, then △ABC is an acute triangle. To understand why the Pythagorean inequalities are true, consider △ABC. If c 2 = a 2 + b 2, then △ABC is a right triangle by the Converse of the Pythagorean Theorem. So m∠C = 90°. If c 2 > a 2 + b 2, then c has increased. By the Converse of the Hinge Theorem, m∠C has also increased. So m∠C > 90°. If c 2 < a 2 + b 2, then c has decreased. By the Converse of the Hinge Theorem, m∠C has also decreased. So m∠C < 90°. E X A M P L E 4 Classifying Triangles By the Triangle Inequality Theorem
, the sum of any two side lengths of a triangle is greater than the third side length. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. A 8, 11, 13 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 8, 11, and 13 can be the side lengths of a triangle. Step 2 Classify the triangle. c 2 ≟ a 2 + b 2 13 2 ≟ 8 2 + 11 2 169 ≟ 64 + 121 169 < 185 Compare c 2 to a 2 + b 2. Substitute the longest side length for c. Multiply. Add and compare. Since c 2 < a 2 + b 2, the triangle is acute. B 5.8, 9.3, 15.6 Step 1 Determine if the measures form a triangle. Since 5.8 + 9.3 = 15.1 and 15.1 ≯ 15.6, these cannot be the side lengths of a triangle. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 4a. 7, 12, 16 4b. 11, 18, 34 4c. 3.8, 4.1, 5.2 351 5- 7 The Pythagorean Theorem 351 �� THINK AND DISCUSS 1. How do you know which numbers to substitute for c, a, and b when using the Pythagorean Inequalities? 2. Explain how the figure at right demonstrates the Pythagorean Theorem. 3. List the conditions that a set of three numbers must satisfy in order to form a Pythagorean triple. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, summarize the Pythagorean relationship. 5-7 Exercises Exercises KEYWORD: MG7 5-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Do the numbers 2.7, 3.6, and 4.5 form a Pythagorean triple? Explain why or why not Find the value of x. Give your answer in simplest radical form. p. 349 2. 3. 4. 349 5. Computers The size of a computer monitor is usually given by the length of its diagonal. A monitor’s aspect ratio is the ratio of its
width to its height. This monitor has a diagonal length of 19 inches and an aspect ratio of 5 : 4. What are the width and height of the monitor? Round to the nearest tenth of an inch. 350 Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 6. 7. 8. 351 Multi-Step Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 9. 7, 10, 12, 1 3 _, 3 1 _ 12. 1 1 _ 4 4 2 10. 9, 11, 15 13. 5.9, 6, 8.4 11. 9, 40, 41 14. 11, 13, 7 √  6 352 352 Chapter 5 Properties and Attributes of Triangles �� PRACTICE AND PROBLEM SOLVING Find the value of x. Give your answer in simplest radical form. 15. 16. 17. 18. Safety The safety rules for a playground state that the height of the slide and the distance from the base of the ladder to the front of the slide must be in a ratio of 3 : 5. If a slide is about 8 feet long, what are the height of the slide and the distance from the base of the ladder to the front of the slide? Round to the nearest inch. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 19. 20. 21. Multi-Step Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 22. 10, 12, 15, 2, 2 1 _ 25. 1 1 _ 2 2 23. 8, 13, 23 26. 0.7, 1.1, 1.7 24. 9, 14, 17 27. 7, 12, 6 √  5 28. Surveying It is believed that surveyors in ancient Egypt laid out right angles using a rope divided into twelve sections by eleven equally spaced knots. How could the surveyors use this rope to make a right angle? 29. /////ERROR ANALYSIS///// Below are two solutions for finding x. Which is incorrect? Explain the error. Independent
Practice For See Exercises Example 15–17 18 19–21 22–27 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 Surveying Ancient Egyptian surveyors were referred to as rope-stretchers. The standard surveying rope was 100 royal cubits. A cubit is 52.4 cm long. Find the value of x. Give your answer in simplest radical form. 30. 31. 33. 34. 32. 35. 353 5- 7 The Pythagorean Theorem 353 �� 36. Space Exploration The International Space Station orbits at an altitude of about 250 miles above Earth’s surface. The radius of Earth is approximately 3963 miles. How far can an astronaut in the space station see to the horizon? Round to the nearest mile. 37. Critical Thinking In the proof of the Pythagorean Theorem on page 348, how do you know the outer figure is a square? How do you know the inner figure is a square? Multi-Step Find the perimeter and the area of each figure. Give your answer in simplest radical form. 38. 41. 39. 42. 40. 43. 44. Write About It When you apply both the Pythagorean Theorem and its converse, you use the equation a 2 + b 2 = c 2. Explain in your own words how the two theorems are different. 45. Use this plan to write a paragraph proof of the Converse of the Pythagorean Theorem. Given: △ABC with a 2 + b 2 = c 2 Prove: △ABC is a right triangle. Plan: Draw △PQR with ∠R as the right angle, leg lengths of a and b, and a hypotenuse of length x. By the Pythagorean Theorem, a 2 + b 2 = x 2. Use substitution to compare x and c. Show that △ABC ≅ △PQR and thus ∠C is a right angle. 46. Complete these steps to prove the Distance Formula. Given: J ( x
1, y 1 ) and K ( x 2, y 2 ) with x 1 ≠ x 2 and y 1 ≠ y 2 Prove: JK = √  ( ̶̶ a. Locate L so that JK is the hypotenuse of right △JKL. What are the coordinates of L? b. Find JL and LK. c. By the Pythagorean Theorem, JK 2 = JL 2 + LK 2. Find JK. 47. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes between four cities. a. A traveler wants to go from Sanak (S) to Manitou (M). To minimize the total number of miles traveled, should she first fly to King City (K) or to Rice Lake (R)? b. The airline decides to offer a direct flight from Sanak (S) to Manitou (M). Given that the length of this flight is more than 1360 mi, what can you say about m∠SRM? 354 354 Chapter 5 Properties and Attributes of Triangles �� 48. Gridded Response ̶̶ KX, ̶̶ LX, and ̶̶̶ MX are the perpendicular bisectors of △GHJ. Find GJ to the nearest tenth of a unit. 49. Which number forms a Pythagorean triple with 24 and 25? 1 7 26 49 50. The lengths of two sides of an obtuse triangle are 7 meters and 9 meters. Which could NOT be the length of the third side? 4 meters 5 meters 11 meters 12 meters 51. Extended Response The figure shows the first six triangles in a pattern of triangles. a. Find PA, PB, PC, PD, PE, and PF in simplest radical form. b. If the pattern continues, what would be the length of the hypotenuse of the ninth triangle? Explain your answer. c. Write a rule for finding the length of the hypotenuse of the nth triangle in the pattern. Explain your answer. CHALLENGE AND EXTEND 52. Algebra
Find all values of k so that (-1, 2), (-10, 5), and (-4, k) are the vertices of a right triangle. 53. Critical Thinking Use a diagram of a right triangle to explain why a + b > √  a 2 + b 2 for any positive numbers a and b. 54. In a right triangle, the leg lengths are a and b, and the length of the altitude to the hypotenuse is h. Write an expression for h in terms of a and b. (Hint: Think of the area of the triangle.) 55. Critical Thinking Suppose the numbers a, b, and c form a Pythagorean triple. Is each of the following also a Pythagorean triple? Explain. a. a + 1, b + 1, c + 1 c. 2a, 2b, 2c d SPIRAL REVIEW Solve each equation. (Previous course) 56. (4 + x) 12 - (4x + 1) 6 = 0 57. 2x - 5 _ 3 = x 58. 4x + 3 (x + 2) = -3 (x + 3) Write a coordinate proof. (Lesson 4-7) 59. Given: ABCD is a rectangle with A (0, 0), B (0, 2b), C (2a, 2b), and D (2a, 0). M is the midpoint of ̶̶ AC. Prove: AM = MB Find the range of values for x. (Lesson 5-6) 60. 61. 355 5- 7 The Pythagorean Theorem 355 �� 5-8 Applying Special Right Triangles TEKS G.5.D Geometric patterns: identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90)... Objectives Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°-60°-90° triangles. Also G.3.B, G.5.A, G.7.A Who uses this? You can use properties of special right triangles to calculate the correct size of a bandana for your dog. (See Example 2.) A diagonal of a square divides it into two congruent isosceles
right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle. A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° triangle √  Pythagorean Theorem Substitute the given values. Simplify. Find the square root of both sides. Simplify. Theorem 5-8-1 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √  2. AC = BC = ℓ AB = Finding Side Lengths in a 45°-45°-90° Triangle Find the value of x. Give your answer in simplest radical form. A By the Triangle Sum Theorem, the measure of the third angle of the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 7. x = 7 √  2 Hypotenuse = leg √  2 356 356 Chapter 5 Properties and Attributes of Triangles �� Find the value of x. Give your answer in simplest radical form. B The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 3. 3 = x √  2 Hypotenuse = leg √  Divide both sides by √  2. Rationalize the denominator. Find the value of x. Give your answer in simplest radical form. 1a. 1b. E X A M P L E 2 Craft Application � � � � Tessa wants to make a bandana for her dog by folding a square of cloth into a 45°-45°-90° triangle. Her dog’s neck has a circumference of about 32 cm. The folded bandana needs to be an extra 16 cm long so Tessa can tie it around her dog’s neck. What should the side length of the square be? Round to the nearest
centimeter. � � �� Tessa needs a 45°-45°-90° triangle with a hypotenuse of 48 cm. 48 = ℓ √  2 ℓ = 48 _ √  2 Divide by √  2 and round. Hypotenuse = leg √  2 ≈ 34 cm 2. What if...? Tessa’s other dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of the other dog’s neck to be? Round to the nearest centimeter. A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths. Draw an altitude in △PQR. Since △PQS ≅ △RQS, ̶̶ PS ≅ the Pythagorean Theorem to find y. ̶̶ RS. Label the side lengths in terms of x, and use 2x ) = √  3 x 2 y = x √  3 Pythagorean Theorem Substitute x for a, y for b, and 2x for c. Multiply and combine like terms. Find the square root of both sides. Simplify. 5- 8 Applying Special Right Triangles 357 357 �� Theorem 5-8-2 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √  3. AC = s AB = 2s BC = Finding Side Lengths in a 30°-60°-90° Triangle Find the values of x and y. Give your answers in simplest radical form. A B 16 = 2x Hypotenuse = 2 (shorter leg) Divide both sides by 2. Longer leg = (shorter leg) √  3 Substitute 8 for x. 11 = x √  3 Longer leg = (shorter leg) √  3 If two angles of a triangle are not congruent, the shorter side lies opposite the smaller angle. = x
= x 11 _ √  3 11 √  3 _ 3 y = 2x y = 2 ( ) _ 11 √  3 3 y = 22 √  3 _ 3 Divide both sides by √  3. Rationalize the denominator. Hypotenuse = 2 (shorter leg) Substitute 11 √  3 _ 3 for x. Simplify. Find the values of x and y. Give your answers in simplest radical form. 3a. 3c. 3b. 3d. 358 358 Chapter 5 Properties and Attributes of Triangles �� 30°-60°-90° Triangles To remember the side relationships in a 30°-60°-90° triangle, I draw a simple “1-2- √  3 ” triangle like this. 2 = 2 (1), so hypotenuse = 2 (shorter leg1), so longer leg = √  3 (shorter leg). Marcus Maiello Johnson High School E X A M P L E 4 Using the 30°-60°-90° Triangle Theorem The frame of the clock shown is an equilateral triangle. The length of one side of the frame is 20 cm. Will the clock fit on a shelf that is 18 cm below the shelf above it? Step 1 Divide the equilateral triangle into two 30°-60°-90° triangles. The height of the frame is the length of the longer leg. Step 2 Find the length x of the shorter leg. Hypotenuse = 2(shorter leg) Divide both sides by 2. 20 = 2x 10 = x Step 3 Find the length h of the longer leg. h = 10 √  3 ≈ 17.3 cm Longer leg = (shorter leg) √  3 The frame is approximately 17.3 centimeters tall. So the clock will fit on the shelf. 4. What if…? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth. THINK AND DISCUSS 1. Explain why an isosceles right triangle is a 45°-45°-90° triangle. 2. Describe how finding x in triangle I is different from finding
x in triangle II. I. II. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, sketch the special right triangle and label its side lengths in terms of s. 5- 8 Applying Special Right Triangles 359 359 �� 5-8 Exercises Exercises GUIDED PRACTICE Find the value of x. Give your answer in simplest radical form. p. 356 1. 2. 3. KEYWORD: MG7 5-8 KEYWORD: MG7 Parent. 357 4. Transportation The two arms of the railroad sign are perpendicular bisectors of each other. In Pennsylvania, the lengths marked in red must be 19.5 inches. What is the distance labeled d? Round to the nearest tenth of an inch Find the values of x and y. Give your answers in simplest radical form. p. 358 5. 6. 7. 359 8. Entertainment Regulation billiard balls are 2 1 __ 4 inches in diameter. The rack used to group 15 billiard balls is in the shape of an equilateral triangle. What is the approximate height of the triangle formed by the rack? Round to the nearest quarter of an inch. Independent Practice For See Exercises Example 9–11 12 13–15 16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING Find the value of x. Give your answer in simplest radical form. 9. 10. 11. 12. Design This tabletop is an isosceles 12. right triangle. The length of the front edge of the table is 48 inches. What is the length w of each side edge? Round to the nearest tenth of an inch. � � �� Find the value of x and y. Give your answers in simplest radical form. 13. 14. 15. 360 360 Chapter 5 Properties and Attributes of Triangles �� Pets 16. Pets A dog walk is used in dog agility 16. competitions. In this dog walk, each ramp makes an angle of 30° with the ground. a. How long is one ramp? b. b. How long is the
entire dog walk, including both ramps? Multi-Step Find the perimeter and area of each figure. Give your answers in simplest radical form. 17. 17. a 45°-45°-90° triangle with hypotenuse length 12 inches 18. 18. a 30°-60°-90° triangle with hypotenuse length 28 centimeters Agility courses test the skill of both the dog and the dog’s handler. Dog agility competitions in the United States are regulated by the U.S. Dog Agility Association, headquartered in Garland, Texas. 19. 19. a square with diagonal length 18 meters 20. an equilateral triangle with side length 4 feet 21. an equilateral triangle with height 30 yards 22. Estimation The triangle loom is made from wood strips shaped into a 45°-45°-90° triangle. Pegs are placed every 1 __ 2 inch along the hypotenuse and every 1 __ 4 inch along each leg. Suppose you make a loom with an 18-inch hypotenuse. Approximately how many pegs will you need? 23. Critical Thinking The angle measures of a triangle are in the ratio 1 : 2 : 3. Are the side lengths also in the ratio 1 : 2 : 3? Explain your answer. Find the coordinates of point P under the given conditions. Give your answers in simplest radical form. 24. △PQR is a 45°-45°-90° triangle with vertices Q (4, 6) and R (-6, -4), and m∠P = 90°. P is in Quadrant II. 25. △PST is a 45°-45°-90° triangle with vertices S (4, -3) and T (-2, 3), and m∠S = 90°. P is in Quadrant I. 26. △PWX is a 30°-60°-90° triangle with vertices W (-1, -4) and X (4, -4), and m∠W = 90°. P is in Quadrant II. 27. △PYZ is a 30°-60°-90° triangle with vertices Y (-7, 10) and Z (5, 10), and m∠Z = 90°. P is in Quadrant IV. 28. Write About It Why do you think 30°-60°-90° triangles and 45°-45°-90° triangles are
called special right triangles? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes among four cities. The airline offers one frequent-flier mile for each mile flown (rounded to the nearest mile). How many frequent-flier miles do you earn for each flight? a. Nelson (N) to Belton (B) b. Idria (I) to Nelson (N) c. Belton (B) to Idria (I) 5- 8 Applying Special Right Triangles 361 361 30°30°12 ft4.5 ftge07se_c05l08007aAB�� 30. Which is a true statement? AB = BC √  2 AB = BC √  3 AC = BC √  3 AC = AB √  2 31. An 18-foot pole is broken during a storm. The top of the pole touches the ground 12 feet from the base of the pole. How tall is the part of the pole left standing? 5 feet 6 feet 13 feet 22 feet �� 32. The length of the hypotenuse of an isosceles right triangle is 24 inches. What is the length of one leg of the triangle, rounded to the nearest tenth of an inch? 13.9 inches 17.0 inches 33.9 inches 41.6 inches 33. Gridded Response Find the area of the rectangle to the nearest tenth of a square inch. CHALLENGE AND EXTEND Multi-Step Find the value of x in each figure. 34. 35. 36. Each edge of the cube has length e. a. Find the diagonal length d when e = 1, e = 2, and e = 3. Give the answers in simplest radical form. b. Write a formula for d for any positive value of e. 37. Write a paragraph proof to show that the altitude to the hypotenuse of a 30°-60°-90° triangle divides the hypotenuse into two segments, one of which is 3 times as long as the other. SPIRAL REVIEW Rewrite each function in the form y = a (x - h) 2 - k and find the axis of symmetry. (Previous course) 38. y = x 2 + 4x 39. y = x 2 - 10x -2 40. y = x 2 + 7x +15 Classify each
triangle by its angle measures. (Lesson 4-1) 41. △ ADB 42. △BDC 43. △ ABC Use the diagram for Exercises 44–46. (Lesson 5-1) 44. Given that PS = SR and m∠PSQ = 65°, find m∠PQR. 45. Given that UT = TV and m∠PQS = 42°, find m∠VTS. 46. Given that ∠PQS ≅ ∠SQR, SR = 3TU, and PS = 7.5, find TV. 362 362 Chapter 5 Properties and Attributes of Triangles �� 5-8 Graph Irrational Numbers Numbers such as √  2 and √  3 are irrational. That is, they cannot be written as the ratio of two integers. In decimal form, they are infinite nonrepeating decimals. You can round the decimal form to estimate the location of these numbers on a number line, or you can use right triangles to construct their locations exactly. Use with Lesson 5-8 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.5.B Activity 1 Draw a line. Mark two points near the left side of the line and label them 0 and 1. The distance from 0 to 1 is 1 unit. 2 Set your compass to 1 unit and mark increments at 2, 3, 4, and 5 units to construct a number line. 3 Construct a perpendicular to the line through 1. 4 Using your compass, mark 1 unit up from the number line and then draw a right triangle. The legs both have length 1, so by the Pythagorean Theorem, the hypotenuse has a length of √2. 5 Set your compass to the length of the hypotenuse. Draw an arc centered at 0 that intersects the number line at √  2. 6 Repeat Steps 3 through 5, starting at √  2, to construct a segment of length √  3. Try This 1. Sketch the two right triangles from Step 6. Label the side lengths and use the Pythagorean Theorem to show why the construction is correct. 2. Construct √4 and verify that
it is equal to 2. 3. Construct √5 through √9 and verify that √9 is equal to 3. 4. Set your compass to the length of the segment from 0 to √2. Mark off another segment of length √2 to show that √8 is equal to 2 √2. 5- 8 Geometry Lab 363 363 �� SECTION 5B Relationships in Triangles Fly Away! A commuter airline serves the four cities of Ashton, Brady, Colfax, and Dumas, located at points A, B, C, and D, respectively. The solid lines in the figure show the airline’s existing routes. The airline is building an airport at H, which will serve as a hub. This will add four new routes to their schedule: ̶̶ CH, and ̶̶ BH, ̶̶ AH, ̶̶̶ DH. 1. The airline wants to locate the airport so that the combined distance to the cities (AH + BH + CH + DH) is as small as possible. Give an indirect argument to explain why the airline should locate the airport at the ̶̶ AC and intersection of the diagonals point X inside quadrilateral ABCD results in a smaller combined distance. Then consider how AX + CX compares to AH + CH.) ̶̶ BD. (Hint: Assume that a different 2. Currently, travelers who want to go from Ashton to Colfax must first fly to Brady. Once the airport is built, they will fly from Ashton to the new airport and then to Colfax. How many miles will this save compared to the distance of the current trip? 3. Currently, travelers who want to go from Brady to Dumas must first fly to Colfax. Once the airport is built, they will fly from Brady to the new airport and then to Dumas. How many miles will this save? 4. Once the airport is built, the airline plans to serve a meal only on its longest flight. On which route should they serve the meal? How do you know that this route is the longest? 364 364 Chapter 5 Properties and Attributes of Triangles �� SECTION 5B Quiz for Lessons 5-5 Through 5-8 5-5 Indirect Proof and Inequalities in One Triangle 1. Write an
indirect proof that the supplement of an acute angle cannot be an acute angle. 2. Write the angles of △KLM in order from smallest to largest. 3. Write the sides of △DEF in order from shortest to longest. Tell whether a triangle can have sides with the given lengths. Explain. 4. 8.3, 10.5, 18.8 5. 4s, s + 10, s 2, when s = 4 6. The distance from Kara’s school to the theater is 9 km. The distance from her school to the zoo is 16 km. If the three locations form a triangle, what is the range of distances from the theater to the zoo? 5-6 Inequalities in Two Triangles 7. Compare PR and SV. 8. Compare m∠KJL and 9. Find the range of m∠MJL. values for x. 5-7 The Pythagorean Theorem 10. Find the value of x. Give the answer in simplest radical form. 11. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 12. Tell if the measures 10, 12, and 16 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 13. A landscaper wants to place a stone walkway from one corner of the rectangular lawn to the opposite corner. What will be the length of the walkway? Round to the nearest inch. 5-8 Applying Special Right Triangles 14. A yield sign is an equilateral triangle with a side length of 36 inches. �� What is the height h of the sign? Round to the nearest inch. Find the values of the variables. Give your answers in simplest radical form. 15. 16. 17. �� Ready to Go On? 365 365 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary altitude of a triangle........ 316 equidistant................. 300 median of a triangle........ 314 centroid of a triangle........ 314 incenter of a triangle
........ 309 midsegment of a triangle.... 322 circumcenter of a triangle... 307 indirect proof............... 332 orthocenter of a triangle.... 316 circumscribed.............. 308 inscribed................... 309 point of concurrency........ 307 concurrent................. 307 locus....................... 300 Pythagorean triple.......... 349 Complete the sentences below with vocabulary words from the list above. 1. A point that is the same distance from two or more objects is? from the objects. ̶̶̶̶ 2. A? is a segment that joins the midpoints of two sides of the triangle. ̶̶̶̶ 3. The point of concurrency of the angle bisectors of a triangle is the?. ̶̶̶̶ 4. A? is a set of points that satisfies a given condition. ̶̶̶̶ 5-1 Perpendicular and Angle Bisectors (pp. 300–306) E X A M P L E S Find each measure. ■ JL ̶̶̶ MK and ̶̶ JM ≅ ̶̶̶ ML is the Because ̶̶ ̶̶̶ ML ⊥ JK, perpendicular bisector of ̶̶ JK. EXERCISES Find each measure. 5. BD 6. YZ TEKS G.3.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B JL = KL ⊥ Bisector Thm. JL = 7.9 Substitute 7.9 for KL. ■ m∠PQS, given that m∠PQR = 68° ̶̶ ̶̶ SP ⊥ QP, Since SP = SR, ̶̶ ̶̶  QS bisects QR, SR ⊥ and ∠PQR by the Converse of the Angle Bisector Theorem.
m∠PQR m∠PQS = 1 _ 2 m∠PQS = 1 _ (68°) = 34° 2 Def. of ∠ bisector Substitute 68° for m∠PQR. 7. HT 8. m∠MNP Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. A (-4, 5), B (6, -5) 10. X (3, 2), Y (5, 10) Tell whether the given information allows you to conclude that P is on the bisector of ∠ABC. 11. 12. 366 366 Chapter 5 Properties and Attributes of Triangles �� 5-2 Bisectors of Triangles (pp. 307–313) TEKS G.2.A, G.2.B, G.3.B, G.7.A, G.7.B E X A M P L E S ̶̶ ̶̶ FG EG, and ■ ̶̶ DG, are the perpendicular bisectors of △ABC. Find AG. G is the circumcenter of △ABC. By the Circumcenter Theorem, G is equidistant from the vertices of △ABC. AG = CG Circumcenter Thm. AG = 5.1 Substitute 5.1 for CG. ■ ̶̶ ̶̶ QS and RS are angle bisectors of △PQR. Find the ̶̶ PR. distance from S to S is the incenter of △PQR. By the Incenter Theorem, S is equidistant from the sides of △PQR. The distance from S to ̶̶ PR is also 17. distance from S to ̶̶ PQ is 17, so the ̶̶ PY, and EXERCISES ̶̶ PX, perpendicular bisectors of △GHJ. Find each length. ̶̶ PZ are the 13. GY 15. GJ 14. GP 16. PH ̶̶ ̶̶ UA and VA are angle bisectors of △UVW. Find each measure. 17. the distance from ̶̶ UV A to 18. m∠WVA Find the circumcenter of a triangle with the given
vertices. 19. M (0, 6), N (8, 0), O (0, 0) 20. O (0, 0), R (0, -7), S (-12, 0) 5-3 Medians and Altitudes of Triangles (pp. 314–320) TEKS G.2.A, G.2.B, G.3.B, G.7.A, G.7.B, G.7.C E X A M P L E S ■ In △JKL, JP = 42. Find JQ. JP JQ = 2_ 3 JQ = 2_ 3 JQ = 28 (42) Centroid Thm. Substitute 42 for JP. Multiply. ■ Find the orthocenter of △RST with vertices R (-5, 3), S (-2, 5), and T (-2, 0). Since ̶̶ ST is vertical, the equation of the line containing the altitude ̶̶ ST is y = 3. from R to ̶̶ RT = slope of 3 - 0 _ -5 - (-2) = -1 ̶̶ RT is 1. The slope of the altitude to This line must pass through S (-2, 5). y - y 1 = m (x - x 1 ) Point-slope form y - 5 = 1 (x + 2) ⎧ y = 3 ⎨ Solve the system y = x + 7 ⎩ Substitution to find that the EXERCISES In △DEF, DB = 24.6, and EZ = 11.6. Find each length. 21. DZ 22. ZB 23. ZC 24. EC Find the orthocenter of a triangle with the given vertices. 25. J (-6, 7), K (-6, 0), L (-11, 0) 26. A (1, 2), B (6, 2), C (1, -8) 27. R (2, 3), S (7, 8), T (8, 3) 28. X (-3, 2), Y (5, 2), Z (3, -4) 29. The coordinates of a triangular piece of a mobile are (0, 4), (3, 8), and (6, 0). The piece will hang from a chain so that
it is balanced. At what coordinates should the chain be attached? coordinates of the orthocenter are (-4, 3). Study Guide: Review 367 367 �� TEKS G.2.A, G.2.B, G.3.B, G.5.A, G.7.B, G.9.B 5-4 The Triangle Midsegment Theorem (pp. 322–327) E X A M P L E S Find each measure. ■ NQ By the △ Midsegment Thm., NQ = 1 _ 2 KL = 45.7. ■ m∠NQM ̶̶̶ ̶̶ ML NP ǁ m∠NQM = m∠PNQ m∠NQM = 37° △ Midsegment Thm. Alt. Int.  Thm. Substitution EXERCISES Find each measure. 30. BC 31. XZ 32. XC 33. m∠BCZ 34. m∠BAX 35. m∠YXZ 36. The vertices of △GHJ are G (-4, -7), H (2, 5), and J (10, -3). V is the midpoint of ̶̶ HJ. Show that W is the midpoint of and VW = 1 __ 2 GJ. ̶̶̶ GH, and ̶̶ ̶̶̶ GJ VW ǁ 5-5 Indirect Proof and Inequalities in One Triangle (pp. 332–339) TEKS G.3.B, E X A M P L E S ■ Write the angles of △RST in order from smallest to largest. EXERCISES 37. Write the sides of △ABC in order from shortest to longest. The smallest angle is opposite the shortest side. In order, the angles are ∠S, ∠R, and ∠T. 38. Write the angles of △FGH in order from smallest to largest. G.3.C, G.3.E, G.5.B ■ The lengths of two sides of a triangle are 15 inches and 12 inches. Find the range of possible lengths for the third side. Let s be the length of the third side. s + 15 > 12 s > -3 s + 12
> 15 s > 3 15 + 12 > s 27 > s By the Triangle Inequality Theorem, 3 in. < s < 27 in. 39. The lengths of two sides of a triangle are 13.5 centimeters and 4.5 centimeters. Find the range of possible lengths for the third side. Tell whether a triangle can have sides with the given lengths. Explain. 40. 6.2, 8.1, 14.2 41. z, z, 3z, when z = 5 42. Write an indirect proof that a triangle cannot have two obtuse angles. 5-6 Inequalities in Two Triangles (pp. 340–345) TEKS G.3.B, G.3.E E X A M P L E S Compare the given measures. ■ KL and ST KJ = RS, JL = RT, and m∠J > m∠R. By the Hinge Theorem, KL > ST. ■ m∠ZXY and m∠XZW XY = WZ, XZ = XZ, and YZ < XW. By the Converse of the Hinge Theorem, m∠ZXY < m∠XZW. 368 368 Chapter 5 Properties and Attributes of Triangles EXERCISES Compare the given measures. 43. PS and RS 44. m∠BCA and m∠DCA Find the range of values for n. 46. 45. �� 5-7 The Pythagorean Theorem (pp. 348–355) TEKS G.1.B, G.5.B, G.5.D, G.8.C, G.11.C E X A M P L E S EXERCISES ■ Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 62 + 32 = x2 45 = x 2 x = 3 √5 Pyth. Thm. Substitution Simplify. Find the positive square root and simplify. ■ Find the missing side length. Tell if the sides form a Pythagorean triple. Explain1.6)2 = 22 a 2 = 1.44 a = 1.2 Pyth. Thm. Substitution Solve for a 2
. Find the positive square root. The side lengths do not form a Pythagorean triple because 1.2 and 1.6 are not whole numbers. Find the value of x. Give your answer in simplest radical form. 47. 48. Find the missing side length. Tell if the sides form a Pythagorean triple. Explain. 49. 50. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 51. 9, 12, 16 52. 11, 14, 27 53. 1.5, 3.6, 3.9 54. 2, 3.7, 4.1 5-8 Applying Special Right Triangles (pp. 356–362) TEKS G.3.B, G.5.A, G.5.D, G.7.A E X A M P L E S EXERCISES Find the values of the variables. Give your answers in simplest radical form. ■ ■ ■ This is a 45°-45°-90° triangle. x = 19 √  2 Hyp. = leg √  2 This is a 45°-45°-90° triangle. 15 = x √  2 Hyp. = leg √  2 Divide both sides by √  2. Rationalize the denominator. This is a 30°-60°-90° triangle. 22 = 2x Hyp. = 2(shorter leg) = x 15 _ √  2 15 √  2 _ 2 = x 11 = x y = 11 √  3 Divide both sides by 2. Longer leg = (shorter leg) √  3 Find the values of the variables. Give your answers in simplest radical form. 55. 56. 57. 59. 58. 60. Find the value of each variable. Round to the nearest inch. 61. 62. Study Guide: Review 369 369 �� Find each measure. 1. KL 2. m∠WXY 3. BC 4. ̶̶̶ NQ, and ̶̶ PQ are the ̶̶̶ MQ, perpendicular bisectors of △RST. Find RS and RQ
. 5. ̶̶ FG are angle ̶̶ EG and bisectors of △DEF. Find m∠GEF and the ̶̶ DF. distance from G to 6. In △XYZ, XC = 261, and ZW = 118. Find XW, BW, and BZ. 7. Find the orthocenter of △JKL with vertices J (-5, 2), K (-5, 10), and L (1, 4). 8. In △GHJ at right, find PR, GJ, and m∠GRP. 9. Write an indirect proof that two obtuse angles cannot form a linear pair. 10. Write the angles of △BEH in order from smallest to largest. 11. Write the sides of △RTY in order from shortest to longest. 12. The distance from Arville to Branton is 114 miles. The distance from Branton to Camford is 247 miles. If the three towns form a triangle, what is the range of distances from Arville to Camford? 13. Compare m∠SPV and m∠ZPV. 14. Find the range of values for x. 15. Find the missing side length in the triangle. Tell if the side lengths form a Pythagorean triple. Explain. 16. Tell if the measures 18, 20, and 27 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 17. An IMAX screen is 62 feet tall and 82 feet wide. What is the length of the screen’s diagonal? Round to the nearest inch. Find the values of the variables. Give your answers in simplest radical form. 18. 19. 20. 370 370 Chapter 5 Properties and Attributes of Triangles �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Some questions on the SAT Mathematics Subject Tests require the use of a calculator. You can take the test without one, but it is not recommended. The calculator you use must meet certain criteria. For example, calculators that make noise or have typewriter-like keypads are not allowed. You may want to time yourself as you take this practice test.
It should take you about 6 minutes to complete. If you have both a scientific and a graphing calculator, bring the graphing calculator to the test. Make sure you spend time getting used to a new calculator before the day of the test. 1. In △ABC, m∠C = 2m∠A, and CB = 3 units. What is AB to the nearest hundredth unit? 3. The side lengths of a right triangle are 2, 5, and c, where c > 5. What is the value of c? (A) 1.73 units (B) 4.24 units (C) 5.20 units (D) 8.49 units (E) 10.39 units 2. What is the perimeter of △ABC if D is the ̶̶ AB, E is the midpoint of ̶̶ BC, and midpoint of F is the midpoint of ̶̶ AC? Note: Figure not drawn to scale. (A) 8 centimeters (B) 14 centimeters (C) 20 centimeters (D) 28 centimeters (E) 35 centimeters (A) √  21 (B) √  29 (C) 7 (D) 9 (E) √  145 4. In the triangle below, which of the following CANNOT be the length of the unknown side? (A) 2.2 (B) 6 (C) 12.8 (D) 17.2 (E) 18.1 5. Which of the following points is on the perpendicular bisector of the segment with endpoints (3, 4) and (9, 4)? (A) (4, 2) (B) (4, 5) (C) (5, 4) (D) (6, -1) (E) (7, 4) College Entrance Exam Practice 371 371 �� Any Question Type: Check with a Different Method It is important to check all of your answers on a test. An effective way to do this is to use a different method to answer the question a second time. If you get the same answer with two different methods, then your answer is probably correct. Multiple Choice What are the coordinates of the centroid of △ABC with A (-2, 4), B (4, 6), and C (1, -1)? (1, 5) (2.5, 2.5) (1,
3) (3, 1) Method 1: The centroid of a triangle is the point of concurrency of the medians. Write the equations of two medians and find their point of intersection. Let D be the midpoint of ̶̶ AB and let E be the midpoint of ̶̶ BC. -1,5) The median from C to D contains C (1, -1) and D (1, 5). 6 + (-12.5, 2.5) It is vertical, so its equation is x = 1. The median from A to E contains A (-2, 4) and E (2.5, 2.5). slope of ̶̶ AE = 4 - 2.5 _ -2 - 2.5 = 1.5 _ -4.x + 2) 3 Point-slope form Substitute 4 for y 1, - 1 _ 3 and -2 for x 1. for m, ⎧ x = 1 Solve the system ⎨ y - 4 = - 1 __ (x + 2) ⎩ 3 to find the point of intersection1 + 2) Substitute 1 for x. y = 3 Simplify. The coordinates of the centroid are (1, 3). So choice H is the correct answer. Method 2: To check this answer, use a different method. By the Centroid Theorem, the centroid of a triangle is 2 __ 3 of the distance from each vertex to the midpoint of the ̶̶ CD is vertical with a length of 6 units. 2 __ (6) = 4, opposite side. 3 and the coordinates of the point that is 4 units up from C is (1, 3). This method confirms that choice H is the correct answer. Problem Solving Strategies • Draw a Diagram • Make a Model • Guess and Test • Work Backward • Find a Pattern • Make a Table • Solve a Simpler Problem • Use Logical Reasoning • Use a Venn Diagram • Make an Organized List 372 372 Chapter 5 Properties and Attributes of Triangles �� If you can’t think of a different method to use to check your answer, circle the question and come back to it later. Item C Gridded Response Find the area of the square in square centimeters. Read each test item and answer the questions that follow. Item A Multiple Choice Given that ℓ is the perpendicular bisector of and BC =
6n - 11, what is the value of n? ̶̶ AB, AC = 3n + 1, 5. How can you use special right triangles to answer this question? 6. Explain how you can check your answer by using the Pythagorean Theorem. -4 3 _ 4 4 _ 3 4 1. How can you use the given answer choices to solve this problem? 2. Describe how to solve this problem directly. Item B Multiple Choice Which number forms a Pythagorean triple with 15 and 17? 5 7 8 10 3. How can you use the given answer choices to find the answer? 4. Describe a different method you can use to check your answer. Item D Multiple Choice Which coordinates for point Z form a right triangle with the points X (-8, 4) and Y (0, -2)? Z (4, 4) Z (4, 6) Z (3, 2) Z (8, 4) 7. Explain how to use slope to determine if △XYZ is a right triangle. 8. How can you use the Converse of the Pythagorean Theorem to check your answer? Item E Multiple Choice What are the coordinates of the orthocenter of △RST? (0, 2) (0, 1) (-1, 3) (1, 2) 9. Describe how you would solve this problem directly. 10. How can you use the third altitude of the triangle to confirm that your answer is correct? TAKS Tackler 373 373 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–5 Multiple Choice 1. ̶̶ GJ is a midsegment of △DEF, and midsegment of △GFJ. What is the length of ̶̶ HK is a ̶̶ HK? 2.25 centimeters 4 centimeters 7.5 centimeters 9 centimeters 2. In △RST, SR < ST, and RT > ST. If m∠R = (2x + 10) ° and m∠T = (3x - 25) °, which is a possible value of x? 25 30 35 40 3. The vertex angle of an isosceles triangle measures (7a - 2) °, and one of the base angles measures (4a + 1) °. Which term best describes this triangle? Acute Equiang
ular Right Obtuse 4. The lengths of two sides of an acute triangle are 8 inches and 10 inches. Which of the following could be the length of the third side? 5 inches 6 inches 12 inches 13 inches 5. For the coordinates M (-1, 0), N (-2, 2), P (10, y), ̶̶̶ MN ǁ ̶̶ PQ. What is the value of y? and Q (4, 6), -18 -6 6 18 6. What is the area of an equilateral triangle that has a perimeter of 18 centimeters? 9 square centimeters 9 √  3 square centimeters 18 square centimeters 18 √  3 square centimeters 7. In △ABC and △DEF, ̶̶ AC ≅ ̶̶ DE, and ∠A ≅ ∠E. Which of the following would allow you to conclude by SAS that these triangles are congruent? ̶̶ AB ≅ ̶̶ AC ≅ ̶̶ BA ≅ ̶̶ CB ≅ ̶̶ DF ̶̶ EF ̶̶ FE ̶̶ DF 8. For the segment below, AB = 1 __ AC, and CD = 2BC. 2 Which expression is equal to the length of ̶̶̶ AD? 2AB + BC 2AC + AB 3AB 4BC 9. In △DEF, m∠D = 2 (m∠E + m∠F). Which term best describes △DEF? Acute Equiangular Right Obtuse 10. Which point of concurrency is always located inside the triangle? The centroid of an obtuse triangle The circumcenter of an obtuse triangle The circumcenter of a right triangle The orthocenter of a right triangle 374 374 Chapter 5 Properties and Attributes of Triangles �� If a diagram is not provided, draw your own. Use the given information to label the diagram. 11. The length of one leg of a right triangle is 3 times the length of the other, and the length of the hypotenuse is 10. What is the length of the longest leg? 3 3 √  10 √  10 12 √  5 12. Which statement is true by the Transitive Property of Congruence? If ∠A ≅ ∠T, then �
�T ≅ ∠A. If m∠L = m∠S, then ∠L ≅ ∠S. 5QR + 10 = 5 (QR + 2) ̶̶ ̶̶ ̶̶ EF, then DE ≅ DE and If ̶̶ BD ≅ ̶̶ BD ≅ ̶̶ EF. Gridded Response 13. P is the incenter of △JKL. The distance from P ̶̶ KL is 2y - 9. What is the distance from P to to ̶̶ JK? STANDARDIZED TEST PREP Short Response 17. In △RST, S is on the perpendicular bisector of m∠S = (4n + 16) °, and m∠R = (3n - 18) °. Find m∠R. Show your work and explain how you determined your answer. ̶̶ RT, 18. Given that ̶̶ BD ǁ ̶̶ AC and ̶̶ AB ≅ ̶̶ BD, explain why AC < DC. 19. Write an indirect proof that an acute triangle cannot contain a pair of complementary angles. Given: △XYZ is an acute triangle. Prove: △XYZ does not contain a pair of complementary angles. 20. Find the coordinates of the orthocenter of △JKL. Show your work and explain how you found your answer. 14. In a plane, r ǁ s, and s ⊥ t. How many right angles are formed by the lines r, s, and t? 15. What is the measure, in degrees, of ∠H? 16. The point T is in the interior of ∠XYZ. If m∠XYZ = (25x + 10) °, m∠XYT = 90°, and m∠TYZ = (9x) °, what is the value of x? Extended Response 21. Consider the statement “If a triangle is equiangular, then it is acute.” a. Write the converse, inverse, and contrapositive of this conditional statement. b. Write a biconditional statement from the conditional statement. c. Determine the truth value of the biconditional statement. If it is false, give a counterexample. d. Determine the truth value of each statement below. Give an example or count
erexample to justify your reasoning. “For any conditional, if the inverse and contrapositive are true, then the biconditional is true.” “For any conditional, if the inverse and converse are true, then the biconditional is true.” Cumulative Assessment, Chapters 1–5 375 375 �� Polygons and Quadrilaterals 6A Polygons and Parallelograms Lab Construct Regular Polygons 6-1 Properties and Attributes of Polygons Lab Explore Properties of Parallelograms 6-2 Properties of Parallelograms 6-3 Conditions for Parallelograms 6B Other Special Quadrilaterals 6-4 Properties of Special Parallelograms Lab Predict Conditions for Special Parallelograms 6-5 Conditions for Special Parallelograms Lab Explore Isosceles Trapezoids 6-6 Properties of Kites and Trapezoids KEYWORD: MG7 ChProj This tile mosaic showing the Alamo and surrounding buildings is on the Riverwalk in San Antonio. 376 376 Chapter 6 Vocabulary Match each term on the left with a definition on the right. 1. exterior angle A. lines that intersect to form right angles 2. parallel lines B. lines in the same plane that do not intersect 3. perpendicular lines C. two angles of a polygon that share a side 4. polygon 5. quadrilateral D. a closed plane figure formed by three or more segments that intersect only at their endpoints E. a four-sided polygon F. an angle formed by one side of a polygon and the extension of a consecutive side Triangle Sum Theorem Find the value of x. 6. 7. 8. 9. Parallel Lines and Transversals Find the measure of each numbered angle. 10. 11. 12. Special Right Triangles Find the value of x. Give the answer in simplest radical form. 13. 15. 14. 16. Conditional Statements Tell whether the given statement is true or false. Write the converse. Tell whether the converse is true or false. 17. If two angles form a linear pair, then they are supplementary. 18. If two angles are congruent, then they are right angles. 19. If a triangle is a scalene triangle, then it is an acute triangle. Polygons and Quadrilaterals 377 377 ��
�� Key Vocabulary/Vocabulario concave diagonal cóncavo diagonal isosceles trapezoid trapecio isósceles kite cometa parallelogram paralelogramo rectangle rectángulo regular polygon polígono regular rhombus square trapezoid rombo cuadrado trapecio Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word concave is made up of two parts: con and cave. Sketch a polygon that looks like it caves in. 2. If a triangle is isosceles, then it has two congruent legs. What do you think is a special property of an isosceles trapezoid? 3. A parallelogram has four sides. What do you think is a special property of the sides of a parallelogram? Geometry TEKS G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships 6-1 Geo. Lab ★ 6-2 Geo. Lab Les. 6-1 Les. 6-2 Les. 6-3 Les. 6-4 6-5 Tech. Lab 6-6 Tech. Lab Les. 6-5 Les. 6-6 ★ ★ ★ ★ ★ ★ G.2.B Geometric structure* make conjectures about... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ polygons... and determine the validity of the conjectures, choosing from a variety of approaches... G.3.B Geometric structure* construct and justify statements ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ about geometric figures and their properties G.3.E Geometric structure* use deductive reasoning to prove ★ ★ ★ ★ ★ a statement G.5.B Geometric patterns* use numeric and geometric ★ ★ patterns to make generalizations about geometric properties, including properties of polygons,... and angle relationships in polygons... G.7.A Dimensionality and the geometry of location* use one- and two-dimensional coordinate systems to represent... figures G.7.B Dimensionality and the geometry of location* use slopes and equations of lines to investigate geometric relationships... G.7.C Dimensionality and the geometry of location
... use formulas involving length, slope, and midpoint ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ G.9.B Congruence and the geometry of size formulate and ★ ★ ★ ★ test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models * Knowledge and skills are written out completely on pages TX28–TX35. 378 378 Chapter 6 Writing Strategy: Write a Convincing Argument Throughout this book, the exercises that require you to write an explanation or argument to support an idea. Your response to a Write About It exercise shows that you have a solid understanding of the mathematical concept. icon identifies To be effective, a written argument should contain • a clear statement of your mathematical claim. • evidence or reasoning that supports your claim. From Lesson 5-4 36. Write About It An isosceles triangle has two congruent sides. Does it also have two congruent midsegments? Explain. Step 1 Make a statement of your mathematical claim. Make a statement of your mathematical claim. Draw a sketch to investigate the properties of the midsegments of an isosceles triangle. You will find that the midsegments parallel to the legs of the isosceles triangle are congruent. Claim: The midsegments parallel to the legs of an isosceles triangle are congruent. Step 2 Give evidence to support your claim. Identify any properties or theorems that support your claim. In this case, the Triangle Midsegment Theorem states that the length of a midsegment of a triangle is 1 __ 2 the length of the parallel side. To clarify your argument, label your diagram and use it in your response. Step 3 Write a complete response. Write a complete response. Yes, the two midsegments parallel to the legs of an isosceles triangle ̶̶ ̶̶ are congruent. Suppose △ABC is isosceles with YZ XZ and are midsegments of △ABC. By the Triangle Midsegment Theorem, ̶̶ ̶̶ AC, AB = AC. So 1 __ 2 AB = 1 __ 2 AC XZ = 1 __ 2 AC and YZ = 1 __ 2 AB. Since AB ≅ by the Multiplication Property of Equality. By substitution, XZ = YZ, so ̶̶ AB ≅ ̶̶ XZ ≅ ̶
̶ AC. ̶̶ YZ. Try This Write a convincing argument. 1. Compare the circumcenter and the incenter of a triangle. 2. If you know the side lengths of a triangle, how do you determine which angle is the largest? Polygons and Quadrilaterals 379 379 6-1 Use with Lesson 6-1 Activity 1 Construct Regular Polygons In Chapter 4, you learned that an equilateral triangle is a triangle with three congruent sides. You also learned that an equilateral triangle is equiangular, meaning that all its angles are congruent. In this lab, you will construct polygons that are both equilateral and equiangular by inscribing them in circles. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures.... Also G.2.B, G.3.B, G.3.D, G.5.B, G.9.B 1 Construct circle P. Draw a diameter ̶̶ AC. 2 Construct the perpendicular bisector of of the bisector and the circle as B and D. ̶̶ AC. Label the intersections 3 Draw ̶̶ AB, ̶̶ BC, ̶̶ CD, and ̶̶ DA. The polygon ABCD is a regular quadrilateral. This means it is a four-sided polygon that has four congruent sides and four congruent angles. Try This 1. Describe a different method for constructing a regular quadrilateral. 2. The regular quadrilateral in Activity 1 is inscribed in the circle. What is the relationship between the circle and the regular quadrilateral? 3. A regular octagon is an eight-sided polygon that has eight congruent sides and eight congruent angles. Use angle bisectors to construct a regular octagon from a regular quadrilateral. Activity 2 1 Construct circle P. Draw a point A on the circle. 2 Use the same compass setting. Starting at A, draw arcs to mark off equal parts along the circle. Label the other points where the arcs intersect the circle as B, C, D, E, and F. 3 Draw ̶̶ BC, ̶̶ CD, ̶̶ AB, ̶̶ FA. The polygon ABCDEF is a regular hexagon. This means it is a six-sided polygon that has six congruent sides and six congruent angles
. ̶̶ EF, and ̶̶ DE, Try This 4. Justify the conclusion that ABCDEF is a regular hexagon. (Hint: Draw ̶̶ AD, ̶̶ BE, and ̶̶ CF. What types of triangles are formed?) diameters 5. A regular dodecagon is a 12-sided polygon that has 12 congruent sides and 12 congruent angles. Use the construction of a regular hexagon to construct a regular dodecagon. Explain your method. 380 380 Chapter 6 Polygons and Quadrilaterals �� Activity 3 1 Construct circle P. Draw a diameter ̶̶ AB. 2 Construct the perpendicular bisector of ̶̶ AB. Label one point where the bisector intersects the circle as point E. 3 Construct the midpoint of radius ̶̶ PB. Label it as point C. 4 Set your compass to the length CE. Place the compass point at C and draw an arc that intersects ̶̶ AB. Label the point of intersection D. 5 Set the compass to the length ED. Starting at E, draw arcs to mark off equal parts along the circle. Label the other points where the arcs intersect the circle as F, G, H, and J. 6 Draw ̶̶ EF, ̶̶ FG, ̶̶̶ GH, ̶̶ JE. The polygon EFGHJ is a regular pentagon. This means it is a five-sided polygon that has five congruent sides and five congruent angles. ̶̶ HJ, and Steps 1–3 Step 4 Step 5 Step 6 Try This 6. A regular decagon is a ten-sided polygon that has ten congruent sides and ten congruent angles. Use the construction of a regular pentagon to construct a regular decagon. Explain your method. 7. Measure each angle of the regular polygons in Activities 1–3 and complete the following table. REGULAR POLYGONS Number of Sides Measure of Each Angle Sum of Angle Measures 3 60° 180° 4 5 6 8. Make a Conjecture What is a general rule for finding the sum of the angle measures in a regular polygon with n sides? 9. Make a Conjecture What is a general rule for finding the measure of each angle in a regular polygon with n sides? 6-1 Geometry Lab 381 381 �� 6-1 Properties and Attributes
of Polygons TEKS G.5.B Geometric patterns: use … patterns to make generalizations about... properties of... and angle relationships in polygons.... Objectives Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Why learn this? The opening that lets light into a camera lens is created by an aperture, a set of blades whose edges may form a polygon. (See Example 5.) Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex Also G.2.B, G.3.B, G.4.A, G.5.A, G.7.A In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons. Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE is a pentagon. Number of Sides Name of Polygon 3 4 5 6 7 8 9 10 12 n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon E X A M P L E 1 Identifying Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. A B C polygon, pentagon not a polygon polygon, octagon Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1a. 1b. 1c. All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. 382 382 Chapter 6 Polygons and Quadrilaterals �� A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is
always convex. E X A M P L E 2 Classifying Polygons Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. A B C irregular, convex regular, convex irregular, concave Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 2a. 2b. To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Polygon Number of Sides Number of Triangles Triangle Quadrilateral Pentagon Hexagon n-gon Sum of Interior Angle Measures (1) 180° = 180° (2) 180° = 360° (3) 180° = 540° (4) 180° = 720° (n - 2) 180° In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n - 2) 180°. Theorem 6-1-1 Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180°. 6- 1 Properties and Attributes of Polygons 383 383 �� E X A M P L E 3 Finding Interior Angle Measures and Sums in Polygons A Find the sum of the interior angle measures of a convex octagon. (n - 2) 180° (8 - 2) 180° 1080° Polygon ∠ Sum Thm. An octagon has 8 sides, so substitute 8 for n. Simplify. B Find the measure of each interior angle of a regular nonagon. Step 1 Find the sum of the interior angle measures. (n - 2) 180° (9 - 2) 180° = 1260° Polygon ∠ Sum Thm. Substitute 9 for n and simplify. Step 2 Find the measure of one interior angle. 1260° _ 9 = 140° The int.  are ≅, so divide by 9. C Find the measure of each interior angle of quadrilateral
PQRS. (4 - 2) 180° = 360° Polygon ∠ Sum Thm. Polygon ∠ Sum Thm. m∠P + m∠Q + m∠R + m∠S = 360° c + 3c + c + 3c = 360 8c = 360 c = 45 Substitute. Combine like terms. Divide both sides by 8. m∠P = m∠R = 45° m∠Q = m∠S = 3 (45°) = 135° 3a. Find the sum of the interior angle measures of a convex 15-gon. 3b. Find the measure of each interior angle of a regular decagon. In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°. An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Theorem 6-1-2 Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. E X A M P L E 4 Finding Exterior Angle Measures in Polygons A Find the measure of each exterior angle of a regular hexagon. A hexagon has 6 sides and 6 vertices. sum of ext.  = 360° measure of one ext. ∠ = 360° _ = 60° 6 Polygon Ext. ∠ Sum Thm. A regular hexagon has 6 ≅ ext. , so divide the sum by 6. The measure of each exterior angle of a regular hexagon is 60°. 384 384 Chapter 6 Polygons and Quadrilaterals �� B Find the value of a in polygon RSTUV. 7a° + 2a° + 3a° + 6a° + 2a° = 360° Polygon Ext. ∠ Sum Thm. 20a = 360 a = 18 Combine like terms. Divide both sides by 20. 4a. Find the measure of each exterior angle of a regular dodecagon. 4b. Find the value of r in polygon JKLM. E X A M P L E 5 Photography Application The aperture of the camera is formed by ten blades. The blades overlap to form a

. of mdpt. 5. △ABC ≅ △EBD 5. ASA Steps 1, 4, 2 17. Given: ̶̶̶ WX ⊥ ̶̶ YZ ⊥ ̶̶̶ WZ ≅ Prove: △WZX ≅ △YXZ ̶̶ XZ, ̶̶ ZX, ̶̶ YX 18. Given: ∠S and ∠V are right angles. RT = UW. m∠T = m∠W Prove: △RST ≅ △UVW 4-6 Triangle Congruence: CPCTC (pp. 260–265) TEKS G.1.A, G.3.E, G.7.A, G.10.B E X A M P L E S ■ Given: ̶̶ JL and Prove: ∠JHG ≅ ∠LKG ̶̶ HK bisect each other. EXERCISES 19. Given: M is the midpoint ̶̶ BD. of ̶̶ BC ≅ ̶̶ DC Prove: ∠1 ≅ ∠2 Proof: Statements ̶̶ HK bisect ̶̶ 1. JL and each other. 2. ̶̶ JG ≅ ̶̶̶ HG ≅ ̶̶ LG, and ̶̶ KG. Reasons 1. Given 2. Def. of bisect 20. Given: ̶̶ RQ, ̶̶ RS ̶̶ PQ ≅ ̶̶ PS ≅ ̶̶ QS bisects ∠PQR. Prove: 3. ∠JGH ≅ ∠LGK 4. △JHG ≅ △LKG 3. Vert.  Thm. 4. SAS Steps 2, 3 5. ∠JHG ≅ ∠LKG 5. CPCTC 286 286 Chapter 4 Triangle Congruence 21. Given: H is the midpoint of L is the midpoint of ̶̶ ̶̶̶ ̶̶̶ GM ≅ KM, GJ ≅ ∠G ≅ ∠K Prove: ∠GMH ≅ ∠KJL ̶̶ KJ, ̶̶ GL. ̶̶̶ MK. �� 4-7

Introduction to Coordinate Proof (pp. 267–272) TEKS G.2.B, G.3.B, G.9.B, G.10.B E X A M P L E S EXERCISES ■ Given: ∠B is a right angle in isosceles right △ABC. E is the midpoint of ̶̶ AB ≅ D is the midpoint of ̶̶ CE ≅ Prove: Proof: Use the coordinates A(0, 2a), B(0, 0), ̶̶ CB. ̶̶ AB. ̶̶ AD ̶̶ CB and C (2a, 0). Draw ̶̶ AD and ̶̶ CE. Position each figure in the coordinate plane and give the coordinates of each vertex. 22. a right triangle with leg lengths r and s 23. a rectangle with length 2p and width p 24. a square with side length 8m For exercises 25 and 26 assign coordinates to each vertex and write a coordinate proof. 25. Given: In rectangle ABCD, E is the midpoint of ̶̶ BC, G is the ̶̶ CD, and H is the midpoint ̶̶ AB, F is the midpoint of midpoint of ̶̶ of AD. ̶̶ EF ≅ ̶̶̶ GH Prove: 26. Given: △PQR has a right ∠Q. ̶̶ PR. M is the midpoint of Prove: MP = MQ = MR 27. Show that a triangle with vertices at (3, 5), (3, 2), and (2, 5) is a right triangle. By the Midpoint Formula, E = ( D = ( 2a + + 2a _ _, 2 2 ) = (0, a) and ) = (a, 0) = (2a - 0) 2 + (0 - a) 2 By the Distance Formula, CE = √  √  4a 2 + a 2 = a √  5 √  (a - 0) 2 + (0 - 2a) 2 √ �

� = ̶̶ AD by the definition of congruence. a 2 + 4a 2 = a √  5 AD = ̶̶ CE ≅ Thus 4-8 Isosceles and Equilateral Triangles (pp. 273–279) TEKS G.2.B, G.3.C, G.10.B E X A M P L E ■ Find the value of x. m∠D + m∠E + m∠F = 180° by the Triangle Sum Theorem. m∠E = m∠F by the Isosceles Triangle Theorem. m∠D + 2 m∠E = 180° Substitution 42 + 2 (3x) = 180 Substitute the given 6x = 138 x = 23 values. Simplify. Divide both sides by 6. EXERCISES Find each value. 28. x 29. RS 30. Given: △ACD is isosceles with ∠D as the vertex angle. B is the midpoint of AB = x + 5, BC = 2x - 3, and CD = 2x + 6. ̶̶ AC. Find the perimeter of △ACD. Study Guide: Review 287 287 �� 1. Classify △ACD by its angle measures. Classify each triangle by its side lengths. 2. △ACD 3. △ABC 4. △ABD 5. While surveying the triangular plot of land shown, a surveyor finds that m∠S = 43°. The measure of ∠RTP is twice that of ∠RTS. What is m∠R? Given: △XYZ ≅ △JKL Identify the congruent corresponding parts. 6. ̶̶ JL ≅? ̶̶̶̶ 10. Given: T is the midpoint of Prove: △PTS ≅ △RTQ 7. ∠Y ≅ ̶̶ PR and? ̶̶̶̶ ̶̶ SQ. 8. ∠L ≅? ̶̶̶̶ ̶̶ YZ ≅ 9.? ̶̶̶̶ 11. The figure represents a walkway with triangular supports. Given that ∠HGK and ∠H ≅ ∠K, use AAS to

prove △HGJ ≅ △KGJ ̶̶ GJ bisects 12. Given: ̶̶ AB ≅ ̶̶ AB ⊥ ̶̶ DC ⊥ Prove: △ABC ≅ △DCB ̶̶ DC, ̶̶ AC, ̶̶ DB 13. Given: Prove: ̶̶ ̶̶ PQ ǁ SR, ∠S ≅ ∠Q ̶̶ ̶̶ QR PS ǁ 14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex. 15. Assign coordinates to each vertex and write a coordinate proof. Given: Square ABCD ̶̶ AC ≅ Prove: ̶̶ BD Find each value. 16. y 17. m∠S 18. Given: Isosceles △ABC has coordinates A (2a, 0), B (0, 2b), and C (-2a, 0). D is the midpoint of ̶̶ AC, and E is the midpoint of ̶̶ AB. Prove: △AED is isosceles. 288 288 Chapter 4 Triangle Congruence �� FOCUS ON ACT The ACT Mathematics Test is one of four tests in the ACT. You are given 60 minutes to answer 60 multiplechoice questions. The questions cover material typically taught through the end of eleventh grade. You will need to know basic formulas but nothing too difficult. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. There is no penalty for guessing on the ACT. If you are unsure of the correct answer, eliminate as many answer choices as possible and make your best guess. Make sure you have entered an answer for every question before time runs out. 1. For the figure below, which of the following must be true? 3. Which of the following best describes a triangle with vertices having coordinates (-1, 0), (0, 3), and (1, -4)? I. m∠EFG > m∠DEF II. m∠EDF = m∠EFD III. m∠DEF + m∠EDF > m∠EFG (A) I only (B) II only (C) I and II

only (D) II and III only (E) I, II, and III 2. In the figure below, △ABD ≅ △CDB, m∠A = (2x + 14) °, m∠C = (3x - 15) °, and m∠DBA = 49°. What is the measure of ∠BDA? (F) 29° (G) 49° (H) 59° (J) 72° (K) 101° (A) Equilateral (B) Isosceles (C) Right (D) Scalene (E) Equiangular 4. In the figure below, what is the value of y? (F) 49 (G) 87 (H) 93 (J) 131 (K) 136 5. In △RST, RS = 2x + 10, ST = 3x - 2, and RT = 1 __ 2 x + 28. If △RST is equiangular, what is the value of x? (A) 2 (B) 5 1 _ 3 (C) 6 (D) 12 (E) 34 College Entrance Exam Practice 289 289 �� Any Question Type: Identify Key Words and Context Clues When reading a test item, you should pay attention to key words and context clues given in the problem statement. These clues will guide you in providing a correct response. Multiple Choice What is the side length of an equilateral triangle with a perimeter of 42 3 __ 4 cm? 42 3 __ 4 cm 24 3 __ 7 cm 21 3 __ 8 cm 14 1 __ 4 cm LOOK for key words and context clues and underline them. Identify what they mean. What is the side length of an equilateral triangle with a perimeter of 42 3 __ 4 in.? equilateral triangle → → perimeter perimeter = 3 (length of one side) a triangle with three congruent sides the distance around a figure = 3 (x) = 3 (x) _ 3 42 3 _ 4 42 3 __ 4 _ 3 171 _ · 1 _ 4 3 14 1 _ 4 = x = x You find the perimeter of an equilateral triangle by multiplying the length of one side of the triangle by three. The correct choice is D because the length of the side of the equilateral triangle is 14 1 __ cm. 4 Gridded Response The vertex angle of an isosceles triangle measures (5t - 5) °, and one

of the base angles measures (t + 5) °. Find t. → isosceles triangle → vertex angle → base angles a triangle with at least two congruent sides the angle formed by the legs The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 2(measure of the base angle) + (measure of the vertex angle) = 180° 2 (t + 5 ) + (5t - 5 ) = 180 2t + 10 + 5t - 5 = 180 7t + 5 = 180 t = 25 The correct value for t is 25. 290 290 Chapter 4 Triangle Congruence �� If you do not understand what a word means, reread the sentences that contain the word and make a logical guess. 6. How will you use the abbreviation SSS to help you answer the question? Read each test item and answer the questions that follow. Item A Multiple Choice Which value of k would make △CDE isosceles. Whether a triangle is isosceles depends on what characteristics of the triangle? 2. What do 2k + 1, 3k - 7, and 5k - 6 represent in the model? 3. How will you use the definition of an isosceles triangle to find the correct value of k? Item C Multiple Choice ∠X and ∠Y are the remote interior angles of ∠YZW in △XYZ. Which of these equations must be true? 180° - m∠X = m∠YZW m∠X = m∠Y + 90° m∠X = m∠YZW - m∠Y m∠YZW = m∠YZX - m∠YXZ 7. Create a drawing that represents the situation. Label the remote interior angles. 8. What is the relationship between the remote interior angles and an exterior angle? 9. How can you manipulate the relationship given in Problem 8 to get one of the four choices? Item D Multiple Choice Which of the following is a correct classification of △FGH? Item B Gridded Response What must the value of x be in order to prove that △MNQ ≅ △PNQ by SSS? Acute Equiangular Isosceles Scalene 4. What statement are you trying to prove? 5. Explain the meaning of the symbol ≅.

10. What are the two ways by which triangles can be classified? 11. What must be true for the triangle to be classified as acute? as equiangular? 12. What must be true for the triangle to be classified as isosceles? as scalene? TAKS Tackler 291 291 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–4 Multiple Choice Use the diagram for Items 1 and 2. 6. Which conditional statement has the same truth value as its inverse? If n < 0, then n 2 > 0. If a triangle has three congruent sides, then it is an isosceles triangle. If an angle measures less than 90°, then it is an acute angle. If n is a negative integer, then n < 0. 1. Which of these congruence statements can be proved from the information given in the figure? △AEB ≅ △CED △ABD ≅ △BCA △BAC ≅ △DAC △DEC ≅ △DEA 7. On a map, an island has coordinates (3, 5), and a reef has coordinates (6, 8). If each map unit represents 1 mile, what is the distance between the island and the reef to the nearest tenth of a mile? 2. What other information is needed to prove that △CEB ≅ △AED by the HL Congruence Theorem? ̶̶̶ AD ≅ ̶̶ BE ≅ ̶̶ AB ̶̶ AE ̶̶ CB ≅ ̶̶ DE ≅ ̶̶̶ AD ̶̶ CE 3. Which biconditional statement is true? Tomorrow is Monday if and only if today is not Saturday. Next month is January if and only if this month is December. Today is a weekend day if and only if yesterday was Friday. This month had 31 days if and only if last month had 30 days. 4. What must be true if   PQ intersects   ST at more than one point? P, Q, S, and T are collinear. P, Q, S, and T are noncoplanar.  PQ and 

 ST are opposite rays.   PQ and   ST are perpendicular. 5. △ABC ≅ △DEF, EF = x 2 - 7, and BC = 4x - 2. Find the values of x. -1 and 5 -1 and 6 1 and 5 2 and 3 292 292 Chapter 4 Triangle Congruence 4.2 miles 6.0 miles 9.0 miles 15.8 miles 8. A line has an x-intercept of -8 and a y-intercept of 3. What is the equation of the line? y = -8x + = 3x - 8 9.   JK passes through points J (1, 3) and K (-3, 11). JK? Which of these lines is perpendicular to   y = -2x - = 2x - 4 10. If PQ = 2 (RS) + 4 and RS = TU + 1, which equation is true by the Substitution Property of Equality? PQ = TU + 5 PQ = TU + 6 PQ = 2 (TU) + 5 PQ = 2 (TU) + 6 11. Which of the following is NOT valid for proving that triangles are congruent? AAA ASA SAS HL �� Use this diagram for Items 12 and 13. STANDARDIZED TEST PREP Short Response 20. Given ℓ ǁ m with transversal n, explain why ∠2 and ∠3 are complementary. 12. What is the measure of ∠ACD? 40° 80° 100° 140° 13. What type of triangle is △ABC? Isosceles acute Equilateral acute Isosceles obtuse Scalene acute �� Take some time to learn the directions for filling in a grid. Check and recheck to make sure you are filling in the grid properly. You will only get credit if the ovals below the boxes are filled in correctly. To check your answer, solve the problem using a different method from the one you originally used. If you made a mistake the first time, you are unlikely to make the same mistake when you solve a different way. Gridded Response 14. △CDE ≅ △JKL. m∠E = (3

x + 4) °, and m∠L = (6x - 5) °. What is the value of x? 15. Lucy, Eduardo, Carmen, and Frank live on the same street. Eduardo’s house is halfway between Lucy’s house and Frank’s house. Lucy’s house is halfway between Carmen’s house and Frank’s house. If the distance between Eduardo’s house and Lucy’s house is 150 ft, what is the distance in feet between Carmen’s house and Eduardo’s house? 16. △JKL ≅ △XYZ, and JK = 10 - 2n. XY = 2, and YZ = n 2. Find KL. 21. ∠G and ∠H are supplementary angles. m∠G = (2x + 12) °, and m∠H = x°. a. Write an equation that can be used to determine the value of x. Solve the equation and justify each step. b. Explain why ∠H has a complement but ∠G does not. 22. A manager conjectures that for every 1000 parts a factory produces, 60 are defective. a. If the factory produces 1500 parts in one day, how many of them can be expected to be defective based on the manager’s conjecture? Explain how you found your answer. b. Use the data in the table below to show that the manager’s conjecture is false. Day Parts Defective Parts 1 2 3 4 5 1000 2000 500 1500 2500 60 150 30 90 150 23. ̶̶ BD is the perpendicular bisector of ̶̶ AC. a. What are the conclusions you can make from this statement? b. Suppose ̶̶ BD intersects is the shortest path from B to ̶̶ AC. ̶̶ AC at D. Explain why ̶̶ BD Extended Response 24. △ABC and △DEF are isosceles triangles. ̶̶ AC ≅ ̶̶ DF. m∠C = 42.5°, and m∠E = 95°. and ̶̶ BC ≅ ̶̶ EF, 17. An angle is its own supplement. What is your answer. a. What is m∠D? Explain how you determined its measure? 18. The area of a circle is 154 square inches. What is its circumference to the nearest inch

that the plane in the diagram has moved along the runway since it passed camera 1? Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List The Great Texas Balloon Race The annual Great Texas Balloon Race is one of the most exciting hot air balloon events in Texas. “Balloon Glow,” in which balloons are tethered and illuminated in an evening display, was begun in Longview, the race’s starting point, in 1980. Traditionally held in July, the race attracts balloonists who compete to fly the obstacle course the most accurately. Choose one or more strategies to solve each problem. 1. The event starts in Longview, and ends near Estes, Texas. The balloons do not fly from the start to the finish in a straight line. They follow a zigzag course to take advantage of the wind. Suppose one of the balloons leaves Longview at a bearing of N 50° E and follows the course shown. At what bearing does the balloon approach Estes? � � �� 2. The speed of the balloon depends on the current wind speed. One event in The Great Texas Balloon Race requires the balloonist to fly to a pole that is 2 mi from the starting point. The balloonist must drop a small ring around the pole, which is 20 ft tall. A second target is 1 mi from the first, a third target is another 3 mi from the second, and a final target is 5 mi farther. If the wind speed is 3.5 mi/h, how long will it take the balloonist to finish the course? Round to the nearest hundredth of an hour. 3. During the race, one of the balloons leaves Longview L, flies to X, and then flies to Y. The team discovers a problem with the balloon, so it must return directly to Longview. Does the table contain enough information to determine the return course to L? Explain to X X to Y Y to L Bearing Distance (mi) N 42° E S 59° E 3.1 2.4 Problem Solving on Location 295 295 Properties and Attributes of Triangles 5A Segments in Triangles 5-1 Perpendicular and Angle Bisectors 5-2 Bisect

ors of Triangles 5-3 Medians and Altitudes of Triangles Lab Special Points in Triangles 5-4 The Triangle Midsegment Theorem 5B Relationships in Triangles Lab Explore Triangle Inequalities 5-5 Indirect Proof and Inequalities in One Triangle 5-6 Inequalities in Two Triangles Lab Hands-on Proof of the Pythagorean Theorem 5-7 The Pythagorean Theorem 5-8 Applying Special Right Triangles Lab Graph Irrational Numbers KEYWORD: MG7 ChProj The Broken Obelisk in Houston was dedicated in 1971 as a memorial to Martin Luther King Jr. 296 296 Chapter 5 Vocabulary Match each term on the left with a definition on the right. 1. angle bisector A. the side opposite the right angle in a right triangle 2. conclusion 3. hypotenuse 4. leg of a right triangle 5. perpendicular bisector of a segment B. a line that is perpendicular to a segment at its midpoint C. the phrase following the word then in a conditional statement D. one of the two sides that form the right angle in a right triangle E. a line or ray that divides an angle into two congruent angles F. the phrase following the word if in a conditional statement Classify Triangles Tell whether each triangle is acute, right, or obtuse. 7. 6. 8. 9. Squares and Square Roots Simplify each expression. 10. 8 2 11. (-12)2 Simplify Radical Expressions Simplify each expression. 14. √9 + 16 15. √ 100 - 36 12. √49 13. - √36 16. √81_ 25 17. √ 2 2 Solve and Graph Inequalities Solve each inequality. Graph the solutions on a number line. 19. -4 ≤ w - 7 18. d + 5 < 1 20. -3s ≥ 6 21. -2 > m_ 10 Logical Reasoning Draw a conclusion from each set of true statements. 22. If two lines intersect, then they are not parallel. Lines ℓ and m intersect at P. 23. If M is the midpoint of ̶̶ AB, then AM = MB. If AM = MB, then AM = 1 __ 2 AB and MB =

1 __ 2 AB. Properties and Attributes of Triangles 297 297 �� Key Vocabulary/Vocabulario altitude of a triangle altura de un triángulo centroid of a triangle centroide de un triángulo circumcenter of a triangle circuncentro de un triángulo concurrent equidistant concurrente equidistante incenter of a triangle incentro de un triángulo median of a triangle mediana de un triángulo midsegment of a triangle segmento medio de un triángulo orthocenter of a triangle orthocentro de un triángulo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. In Latin, co means “together with,” and currere means “to run.” How can you use these meanings to understand what concurrent lines are? 2. The endpoints of a midsegment of a triangle are on two sides of the triangle. Where on the sides do you think the endpoints are located? 3. The strip of concrete or grass in the middle of some roadways is called the median. What do you think the term median of a triangle means? 4. Think of the everyday meaning of altitude. What do you think the altitude of a triangle is? Geometry TEKS Les. 5-1 Les. 5-2 Les. 5-3 5-3 Tech. Lab 5-5 Geo. Lab Les. 5-4 Les. 5-5 Les. 5-6 5-7 Geo. Lab Les. 5-7 Les. 5-8 G.2.A Geometric structure* use constructions to ★ ★ ★ ★ explore attributes of geometric figures and to make conjectures... G.3.B Geometric structure* construct and justify ★ ★ ★ ★ ★ ★ ★ 5-8 Geo. Lab ★ statements about geometric figures and their properties G.5.D Geometric patterns* identify and apply patterns from right triangles... including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples G.7.B Dimensionality and the geometry of location*... investigate geometric relationships, including... special segments of triangles... G.8.C Congruence and the geometry of size* derive, extend

, and use the Pythagorean Theorem G.9.B Congruence and the geometry of size* formulate and test conjectures about... polygons and their component parts... G.11.C Similarity and the geometry of shape* develop, apply, and justify triangle similarity relationships, such as... Pythagorean triples... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 298 298 Chapter 5 Reading Strategy: Learn Math Vocabulary Mathematics has a vocabulary all its own. To learn and remember new vocabulary words, use the following study strategies. • Try to figure out the meaning of a new word based on its context. • Use a dictionary to look up the root word or prefix. • Relate the new word to familiar everyday words. Once you know what a word means, write its definition in your own words. Term Study Notes Definition Polygon The prefix poly means “many” or “several.” Bisect Slope Intersection The prefix bi means “two.” Think of a ski slope. The root word intersect means “to overlap.” Think of the intersection of two roads. A closed plane figure formed by three or more line segments Cuts or divides something into two equal parts The measure of the steepness of a line The set of points that two or more lines have in common Try This Complete the table below. Term Study Notes Definition 1. Trinomial 2. Equiangular triangle 3. Perimeter 4. Deductive reasoning Use the given prefix and its meanings to write a definition for each vocabulary word. 5. circum (about, around); circumference 6. co (with, together); coplanar 7. trans (across, beyond, through); translation Properties and Attributes of Triangles 299 299 5-1 Perpendicular and Angle Bisectors TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.E, G.7.A, G.7.B, G.7.C, G.10.B Objectives Prove and apply theorems about perpendicular bisectors. Prove and apply theorems about angle bisectors. Vocabulary equidistant locus Who uses this? The suspension and steering lines of a parachute keep the sky diver centered under the parachute. (See Example 3.) When a point is the same distance from

two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points. Theorems Distance and Perpendicular Bisectors THEOREM HYPOTHESIS CONCLUSION 5-1-1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 5-1-2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. XA = XB ̶̶ XY ⊥ ̶̶ YA ≅ ̶̶ AB ̶̶ YB ̶̶ XY ⊥ ̶̶ YA ≅ ̶̶ AB ̶̶ YB XA = XB You will prove Theorem 5-1-2 in Exercise 30. PROOF PROOF Perpendicular Bisector Theorem Given: ℓ is the perpendicular bisector of Prove: XA = XB ̶̶ AB. The word locus comes from the Latin word for location. The plural of locus is loci, which is pronounced LOW-sigh. Proof: ̶̶ AB, ℓ ⊥ ̶̶ AB and Y is the midpoint ̶̶ AB. By the definition of perpendicular, ∠AYX and ∠BYX are right ̶̶ BY. Since ℓ is the perpendicular bisector of of angles and ∠AYX ≅ ∠BYX. By the definition of midpoint, By the Reflexive Property of Congruence, by SAS, and of congruent segments. ̶̶ XY. So △AYX ≅ △BYX ̶̶ XB by CPCTC. Therefore XA = XB by the definition ̶̶ XA ≅ ̶̶ XY ≅ ̶̶ AY ≅ A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. 300 300 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Applying the Perpendicular Bisector The

orem and Its Converse Find each measure. A YW YW = XW YW = 7.3 B BC ⊥ Bisector Thm. Substitute 7.3 for XW. ̶̶ BC, ℓ is the perpendicular Since AB = AC and ℓ ⊥ ̶̶ BC by the Converse of the bisector of Perpendicular Bisector Theorem. BC = 2CD BC = 2 (16) = 32 Def. of seg. bisector Substitute 16 for CD. C PR PR = RQ 2n + 9 = 7n - 18 9 = 5n - 18 27 = 5n 5.4 = n ⊥ Bisector Thm. Substitute the given values. Subtract 2n from both sides. Add 18 to both sides. Divide both sides by 5. So PR = 2 (5.4) + 9 = 19.8. Find each measure. 1a. Given that line ℓ is the perpendicular ̶̶ DE and EG = 14.6, find DG. bisector of 1b. Given that DE = 20.8, DG = 36.4, and EG = 36.4, find EF. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. Theorems Distance and Angle Bisectors THEOREM HYPOTHESIS CONCLUSION 5-1-3 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. 5-1-4 Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. ∠APC ≅ ∠BPC AC = BC ∠APC ≅ ∠BPC AC = BC You will prove these theorems in Exercises 31 and 40. 5- 1 Perpendicular and Angle Bisectors 301 301 �� Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle. E X A M P L E 2 Applying the Angle Bisector Theorems Find each measure. A LM LM = JM LM = 12

.8 ∠ Bisector Thm. Substitute 12.8 for JM. B m∠ABD, given that m∠ABC = 112° ̶̶ BA, and ̶̶ AD ⊥  BD bisects ∠ABC Since AD = DC, ̶̶ ̶̶ DC ⊥ BC, by the Converse of the Angle Bisector Theorem. m∠ABD = 1 _ 2 m∠ABD = 1 _ (112°) = 56° Substitute 112° for m∠ABC. 2 Def. of ∠ bisector m∠ABC C m∠TSU ̶̶ UT ⊥ ̶̶ SR, and ̶̶ ̶̶ Since RU = UT, ST, RU ⊥  SU bisects ∠RST by the Converse of the Angle Bisector Theorem. m∠RSU = m∠TSU 6z + 14 = 5z + 23 z + 14 = 23 Def. of ∠ bisector Substitute the given values. Subtract 5z from both sides. z = 9 Subtract 14 from both sides. ⎤ ⎦ ⎡ ⎣ ° = 68°. 5 (9) + 23 So m∠TSU = Find each measure. 2a. Given that  YW bisects ∠XYZ and WZ = 3.05, find WX. 2b. Given that m∠WYZ = 63°, XW = 5.7, and ZW = 5.7, find m∠XYZ. E X A M P L E 3 Parachute Application Each pair of suspension lines on a parachute are the same length and are equally spaced from the center of the chute. How do these lines keep the sky diver centered under the parachute? ̶̶ PQ ≅ ̶̶ RQ. So Q is It is given that on the perpendicular bisector ̶̶ PR by the Converse of the of Perpendicular Bisector Theorem. Since S is the midpoint of ̶̶ QS is the perpendicular bisector ̶̶ PR. Therefore the sky diver of remains centered under the chute. ̶̶ PR, 302 302 Chapter 5 Properties and Attributes of Triangles ��

�� 3. S is equidistant from each pair of suspension lines. What can you conclude about  QS? E X A M P L E 4 Writing Equations of Bisectors in the Coordinate Plane Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints A (-1, 6) and B (3, 4). ̶̶ AB. Step 1 Graph The perpendicular bisector of perpendicular to ̶̶ AB is ̶̶ AB at its midpoint. ̶̶ AB. Step 2 Find the midpoint of AB = ( 6 + 4 -1 + 3 _ _, 2 2 mdpt. of ̶̶ ) = (1, 5) Midpoint formula Step 3 Find the slope of the perpendicular bisector. slope = slope of ̶̶ AB = 4 - 6 _ 3 - (-1) Slope formula = -2 _ = - 1 _ 4 2 Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is 2. Step 4 Use point-slope form to write an equation. The perpendicular bisector of ̶̶ AB has slope 2 and passes through (1, 5). y - y 1 = m (x - 1 ) Point-slope form Substitute 5 for y 1, 2 for m, and 1 for x 1. 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P (5, 2) and Q (1, -4). THINK AND DISCUSS ̶̶ PQ? Is it a perpendicular 1. Is line ℓ a bisector of bisector of ̶̶ PQ? Explain. 2. Suppose that M is in the interior of ∠JKL and MJ = ML. Can you conclude that is the bisector of ∠JKL? Explain.  KM 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the theorem or its converse in your own words. 5- 1 Perpendicular and Angle Bisectors 303 303 �� 5-1 Exercises Exercises KEYWORD: MG7 5-1 KEYWORD: MG7 Parent GUIDED PRACTICE

1. Vocabulary A from the endpoints of a segment. (perpendicular bisector or angle bisector)? is the locus of all points in a plane that are equidistant ̶̶̶̶ Use the diagram for Exercises 2–4. p. 301 2. Given that PS = 53.4, QT = 47.7, and QS = 53.4, find PQ. 3. Given that m is the perpendicular bisector ̶̶ PQ and SQ = 25.9, find SP. of 4. Given that m is the perpendicular bisector ̶̶ PQ, PS = 4a, and QS = 2a + 26, find QS. of Use the diagram for Exercises 5–7. p. 302 5. Given that  BD bisects ∠ABC and CD = 21.9, find AD. 6. Given that AD = 61, CD = 61, and m∠ABC = 48°, find m∠CBD. 7. Given that DA = DC, m∠DBC = (10y + 3) °, and m∠DBA = (8y + 10) °, find m∠DBC. 302 8. Carpentry For a king post truss to be constructed correctly, P must lie on the bisector of ∠JLN. How can braces and the proper location? ̶̶ PK ̶̶̶ PM be used to ensure that P is in. 303 Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. M (-5, 4), N (1, -2) 10. U (2, -6), V (4, 0) 11. J (-7, 5), K (1, -1) Independent Practice Use the diagram for Exercises 12–14. PRACTICE AND PROBLEM SOLVING For See Exercises Example 12–14 15–17 18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 12. Given that line t is the perpendicular bisector ̶̶ JK and GK = 8.25, find GJ. of 13. Given that line t is the perpendicular bisector ̶̶ JK, JG = x +

12, and KG = 3x - 17, find KG. of 14. Given that GJ = 70.2, JH = 26.5, and GK = 70.2, find JK. Use the diagram for Exercises 15–17. 15. Given that m∠RSQ = m∠TSQ and TQ = 1.3, find RQ. 16. Given that m∠RSQ = 58°, RQ = 49, and TQ = 49, find m∠RST. 17. Given that RQ = TQ, m∠QSR = (9a + 48) °, and m∠QST = (6a + 50) °, find m∠QST. 304 304 Chapter 5 Properties and Attributes of Triangles ��ge07sec05l01003aa1st pass4/4/5cmurphyLMNKJP�� 18. City Planning The planners for a new section of the city want every location on Main Street to be equidistant from Elm Street and Grove Street. How can the planners ensure that this is the case? Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 19. E (-4, -7), F (0, 1) 20. X (-7, 5), Y (-1, -1) ̶̶ ST. 22. ̶̶ PQ is the perpendicular bisector of Find the values of m and n. 21. M (-3, -1), N (7, -5) Shuffleboard One of the first recorded shuffleboard games was played in England in 1532. In this game, Henry VIII supposedly lost £9 to Lord William. Shuffleboard Use the diagram of a shuffleboard and the following information to find each length in Exercises 23–28. ̶̶ KZ is the perpendicular bisector of ̶̶̶ HM, and ̶̶̶ GN, ̶̶ JL. 23. JK 24. GN 25. ML 27. JL 26. HY 29. Multi-Step The endpoints of 28. NM ̶̶ AB are A (-2, 1) and B (4, -3). Find the coordinates of a point C other than the midpoint of you know it is on the perpendicular bisector? � ��

�� ̶̶ AB that is on the perpendicular bisector of ̶̶ AB. How do 30. Write a paragraph proof of the Converse of the Perpendicular Bisector Theorem. Given: AX = BX Prove: X is on the perpendicular bisector of ̶̶ AB. Plan: Draw ℓ perpendicular to ̶̶ △AYX ≅ △BYX and thus AY ≅ the perpendicular bisector of ̶̶ AB through X. Show that ̶̶ BY. By definition, ℓ is ̶̶ AB. 31. Write a two-column proof of the Angle Bisector Theorem. ̶̶ SQ ⊥  PS bisects ∠QPR. ̶̶ SR ⊥  PQ,  PR Given: Prove: SQ = SR Plan: Use the definitions of angle bisector and perpendicular to identify two pairs of congruent angles. Show that △PQS ≅ △PRS and thus ̶̶ SQ ≅ ̶̶ SR. 32. Critical Thinking In the Converse of the Angle Bisector Theorem, why is it important to say that the point must be in the interior of the angle? 33. This problem will prepare you for the Multi-Step TAKS Prep on page 328. A music company has stores in Abby (-3, -2) and Cardenas (3, 6). Each unit in the coordinate plane represents 1 mile. a. The company president wants to build a warehouse that is equidistant from the two stores. Write an equation that describes the possible locations. b. A straight road connects Abby and Cardenas. The warehouse will be located exactly 4 miles from the road. How many locations are possible? c. To the nearest tenth of a mile, how far will the warehouse be from each store? 5- 1 Perpendicular and Angle Bisectors 305 305 Elm StreetMain StreetGrove StreetHolt, Rinehart & WinstonGeometry 2007ge07se_c05l01004a 1st proofCity Planning map�� 34. Write About It How is the construction of the perpendicular bisector of a segment related to the Converse of the Perpendicular Bisector Theorem? 35.

If   JK is perpendicular to JX = KY ̶̶ XY at its midpoint M, which statement is true? JX = KX JM = KM JX = JY 36. What information is needed to conclude that m∠DEF = m∠DEG m∠FEG = m∠DEF  EF is the bisector of ∠DEG? m∠GED = m∠GEF m∠DEF = m∠EFG 37. Short Response The city wants to build a visitor center in the park so that it is equidistant from Park Street and Washington Avenue. They also want the visitor center to be equidistant from the museum and the library. Find the point V where the visitor center should be built. Explain your answer. CHALLENGE AND EXTEND 38. Consider the points P (2, 0), A (-4, 2), B (0, -6), and C (6, -3). a. Show that P is on the bisector of ∠ABC. b. Write an equation of the line that contains the bisector of ∠ABC. 39. Find the locus of points that are equidistant from the x-axis and y-axis. 40. Write a two-column proof of the Converse of the Angle Bisector Theorem. Given: Prove: ̶̶ ̶̶ VZ ⊥  YX, VX ⊥  YV bisects ∠XYZ.  YZ, VX = VZ 41. Write a paragraph proof. Given: ̶̶ KN is the perpendicular bisector of ̶̶ LN is the perpendicular bisector of ̶̶ JR ≅ Prove: ∠JKM ≅ ∠MLJ ̶̶̶ MT ̶̶ JL. ̶̶̶ KM. SPIRAL REVIEW 42. Lyn bought a sweater for $16.95. The change c that she received can be described by c = t - 16.95, where t is the amount of money Lyn gave the cashier. What is the dependent variable? (Previous course) For the points R (-4, 2), S (1, 4), T

(3, -1), and V (-7, -5), determine whether the lines are parallel, perpendicular, or neither. (Lesson 3-5) 43.   RS and   VT 44.   RV and   ST 45.   RT and   VR Write the equation of each line in slope-intercept form. (Lesson 3-6) 46. the line through the points (1, -1) and (2, -9) 47. the line with slope -0.5 through (10, -15) 48. the line with x-intercept -4 and y-intercept 5 306 306 Chapter 5 Properties and Attributes of Triangles ge07sec05l01006a3rd Pass2/24/05C MurphyMuseumPark StreetWashington AvenueLibrary�� 5-2 Bisectors of Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.A, G.7.B Objectives Prove and apply properties of perpendicular bisectors of a triangle. Prove and apply properties of angle bisectors of a triangle. Vocabulary concurrent point of concurrency circumcenter of a triangle circumscribed incenter of a triangle inscribed The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex. Who uses this? An event planner can use perpendicular bisectors of triangles to find the best location for a fireworks display. (See Example 4.) Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property. Construction Circumcenter of a Triangle    Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point

of concurrency is the circumcenter of the triangle. Theorem 5-2-1 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC The circumcenter can be inside the triangle, outside the triangle, or on the triangle. 5- 2 Bisectors of Triangles 307 307 �� The circumcenter of △ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. PROOF PROOF Circumcenter Theorem Given: Lines ℓ, m, and n are the perpendicular ̶̶ ̶̶ AC, respectively. BC, and ̶̶ AB, bisectors of Prove: PA = PB = PC Proof: P is the circumcenter of △ABC. Since P lies on the perpendicular bisector of by the Perpendicular Bisector Theorem. Similarly, P also lies on the perpendicular bisector of by the Transitive Property of Equality. ̶̶ BC, so PB = PC. Therefore PA = PB = PC ̶̶ AB, PA = PB E X A M P L E 1 Using Properties of Perpendicular Bisectors ̶̶ LZ, and ̶̶ KZ, of △GHJ. Find HZ. ̶̶ MZ are the perpendicular bisectors Z is the circumcenter of △GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of △GHJ. HZ = GZ HZ = 19.9 Circumcenter Thm. Substitute 19.9 for GZ. Use the diagram above. Find each length. 1a. GM 1b. GK 1c. JZ E X A M P L E 2 Finding the Circumcenter of a Triangle Find the circumcenter of △RSO with vertices R (-6, 0), S (0, 4), and O (0, 0). Step 1 Graph the triangle. Step 2 Find equations for two perpendicular bisectors. Since two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of x = -3, and the perpendicular bisector of ̶̶ OS is y = 2. ̶̶ RO is Step 3

Find the intersection of the two equations. The lines x = -3 and y = 2 intersect at (-3, 2 ), the circumcenter of △RSO. 308 308 Chapter 5 Properties and Attributes of Triangles �� 2. Find the circumcenter of △GOH with vertices G (0, -9), O (0, 0), and H (8, 0). A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle. Theorem 5-2-2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ You will prove Theorem 5-2-2 in Exercise 35. Unlike the circumcenter, the incenter is always inside the triangle. The distance between a point and a line is the length of the perpendicular segment from the point to the line. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point. E X A M P L E 3 Using Properties of Angle Bisectors ̶̶ KV are angle bisectors of △JKL. ̶̶ JV and Find each measure. A the distance from V to ̶̶ KL V is the incenter of △JKL. By the Incenter Theorem, V is equidistant from the sides of △JKL. The distance from V to So the distance from V to ̶̶ JK is 7.3. ̶̶ KL is also 7.3. B m∠VKL m∠KJL = 2m∠VJL m∠KJL = 2 (19°) = 38° m∠KJL + m∠JLK + m∠JKL = 180° 38 + 106 + m∠JKL = 180 m∠JKL = 36° m∠JKL m∠VKL = 1 _ 2 m∠VKL = 1 _ (36°) = 18° 2 ̶̶ JV is the bisector of ∠KJL. Substitute 19° for m�

�VJL. △ Sum Thm. Substitute the given values. Subtract 144° from both sides. ̶̶ KV is the bisector of ∠JKL. Substitute 36° for m∠JKL. 5- 2 Bisectors of Triangles 309 309 �� ̶̶ RX are angle bisectors ̶̶ QX and of △PQR. Find each measure. ̶̶ PQ 3a. the distance from X to 3b. m∠PQX E X A M P L E 4 Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing locations A, B, and C on the map. Justify your sketch. Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. 4. A city plans to build a � � � � firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch. � � � � � � � � �� THINK AND DISCUSS 1. Sketch three lines that are concurrent. 2. P and Q are the circumcenter and incenter of △RST, but not necessarily in that order. Which point is the circumcenter? Which point is the incenter? Explain how you can tell without constructing any of the bisectors. 3. GET ORGANIZED Copy and complete the graphic organizer. Fill in the blanks to make each statement true. 310 310 Chapter 5 Properties and Attributes of Triangles BAC38520print-ready file 7/12/05ge07ts_c05l02005aGeometry SE 2007 Texasmap of OdessaHolt Rinehart WinstonKaren Minot(415)883

-6560�� 5-2 Exercises Exercises KEYWORD: MG7 5-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain why lines ℓ, m, and n are NOT concurrent. 2. A circle that contains all the vertices of a polygon is. 308? the polygon. (circumscribed about or inscribed in) ̶̶̶̶ ̶̶ TN, and ̶̶ VN are the perpendicular bisectors ̶̶ SN, of △PQR. Find each length. 3. NR 5. TR 4. RV 6. QN Multi-Step Find the circumcenter of a triangle with the given vertices. p. 308 7. O (0, 0), K (0, 12), L (4, 0. 309 8. A (-7, 0), O (0, 0), B (0, -10) ̶̶ CF and Find each measure. ̶̶ EF are angle bisectors of △CDE. 9. the distance from F to ̶̶ CD 10. m∠FED 11. Design The designer of the p. 310 Newtown High School pennant wants the circle around the bear emblem to be as large as possible. Draw a sketch to show where the center of the circle should be located. Justify your sketch. Independent Practice For See Exercises Example 12–15 16–17 18–19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING ̶̶ DY, of △ABC. Find each length. ̶̶ FY are the perpendicular bisectors ̶̶ EY, and 12. CF 14. DB 13. YC 15. AY Multi-Step Find the circumcenter of a triangle with the given vertices. 16. M (-5, 0), N (0, 14), O (0, 0)

17. O (0, 0), V (0, 19), W (-3, 0) ̶̶ SJ are angle bisectors of △RST. ̶̶ TJ and Find each measure. 18. the distance from J to ̶̶ RS 19. m∠RTJ 5- 2 Bisectors of Triangles 311 311 �� 20. Business A company repairs photocopiers in Harbury, Gaspar, and Knowlton. Draw a sketch to show where the company should locate its office so that it is the same distance from each city. Justify your sketch. 21. Critical Thinking If M is the incenter of △JKL, explain why ∠JML cannot be a right angle. Tell whether each segment lies on a perpendicular bisector, an angle bisector, or neither. Justify your answer. 22. 25. ̶̶ AE ̶̶ CR 23. 26. ̶̶̶ DG ̶̶ FR 24. 27. ̶̶ BG ̶̶ DR Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 28. The angle bisectors of a triangle intersect at a point outside the triangle. 29. An angle bisector of a triangle bisects the opposite side. 30. A perpendicular bisector of a triangle passes through the opposite vertex. 31. The incenter of a right triangle is on the triangle. 32. The circumcenter of a scalene triangle is inside the triangle. Algebra Find the circumcenter of the triangle with the given vertices. 33. O (0, 0), A (4, 8), B (8, 0) 34. O (0, 0), Y (0, 12), Z (6, 6) 35. Complete this proof of the Incenter Theorem by filling in the blanks. Given:  AP, ̶̶ PX ⊥ Prove: PX = PY = PZ  BP, and ̶̶ ̶̶ PY ⊥ AC, ̶̶ AB, ̶̶ PZ ⊥ ̶̶ BC  CP bisect ∠A, ∠B, and ∠C, respectively. Proof:

Let P be the incenter of △ABC. Since P lies on the bisector of ∠A, PX = PY by a. Similarly, P also lies on b. Therefore c.?. ̶̶̶̶?, so PY = PZ. ̶̶̶̶? by the Transitive Property of Equality. ̶̶̶̶ 36. Prove that the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Given:   QS bisects ∠PQR. Prove:   QS is the perpendicular bisector of ̶̶ PR. ̶̶ PQ ≅ ̶̶ RQ Plan: Show that △PQS ≅ △RQS. Then use CPCTC to show that S is the midpoint of ̶̶ PR and that   QS ⊥ ̶̶ PR. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 328. A music company has stores at A (0, 0), B (8, 0), and C (4, 3), where each unit of the coordinate plane represents one mile. a. A new store will be built so that it is equidistant from the three existing stores. Find the coordinates of the new store’s location. b. Where will the new store be located in relation to △ABC? c. To the nearest tenth of a mile, how far will the new store be from each of the existing stores? 312 312 Chapter 5 Properties and Attributes of Triangles HarburyGasparKnowltonHarburyGasparKnowltonHolt, Rinehart & WinstonGeometry 2007ge07se_c05l02008a 1st pass3/9/5C MurphyBisector map�� 38. Write About It How are the inscribed circle and the circumscribed circle of a triangle alike? How are they different? 39. Construction Draw a large scalene acute triangle. a. Construct the angle bisectors to find the incenter. Inscribe a circle in the triangle. b. Construct the perpendicular bisectors to find the circumcenter. Circumscribe a circle around the triangle. 40. P is the incenter of △ABC.

Which must be true? PA = PB PX = PY YA = YB AX = BZ 41. Lines r, s, and t are concurrent. The equation of line r is x = 5, and the equation of line s is y = -2. Which could be the equation of line t 42. Gridded Response Lines a, b, and c are the perpendicular bisectors of △KLM. Find LN. CHALLENGE AND EXTEND 43. Use the right triangle with the given coordinates. a. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. b. Make a conjecture about the circumcenter of a right triangle. 44. Design A trefoil is created by constructing a circle at each vertex of an equilateral triangle. The radius of each circle equals the distance from each vertex to the circumcenter of the triangle. If the distance from one vertex to the circumcenter is 14 cm, what is the distance AB across the trefoil? Design The trefoil shape, as seen in this stained glass window, has been used in design for centuries. SPIRAL REVIEW Solve each proportion. (Previous course) 45. t _ 26 46. 2.5 _ 1.75 = 10 _ 65 = 6 _ x 47. 420 _ y = 7 _ 2 Find each angle measure. (Lesson 2-6) 48. m∠BFE 49. m∠BFC 50. m∠CFE Determine whether each point is on the perpendicular bisector of the segment with endpoints S (0, 8) and T (4, 0). (Lesson 5-1) 51. X (0, 3) 53. Z (-8, -2) 52. Y (-4, 1) 5- 2 Bisectors of Triangles 313 313 �� 5-3 Medians and Altitudes of Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.A, G.7.B, G.7.C Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle. Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a

triangle Who uses this? Sculptors who create mobiles of moving objects can use centers of gravity to balance the objects. (See Example 2.) A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent, as shown in the construction below. Construction Centroid of a Triangle    Draw △ABC. Construct the ̶̶ midpoints of BC, and Label the midpoints of the sides X, Y, and Z, respectively. ̶̶ AB, ̶̶ AC. ̶̶ AY, ̶̶ Draw BZ, and the three medians of △ABC. ̶̶ CX. These are Label the point where and ̶̶ CX intersect as P. ̶̶ AY, ̶̶ BZ, The point of concurrency of the medians of a triangle is the centroid of the triangle. The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. Theorem 5-3-1 Centroid Theorem The centroid of a triangle is located 2 __ of the distance 3 from each vertex to the midpoint of the opposite side. AP = 2 _ 3 AY BP = 2 _ 3 BZ CP = 2 _ 3 CX 314 314 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Centroid to Find Segment Lengths In △ABC, AF = 9, and GE = 2.4. Find each length. A AG AF AG = 2 _ 3 AG = 2 _ (9) 3 AG = 6 B CE CG = 2 _ 3 CE Centroid Thm. Substitute 9 for AF. Simplify. Centroid Thm. CG + GE = CE 2 _ CE + GE = CE 3 GE = 1 _ 3 2.4 = 1 _ 3 CE CE Seg. Add. Post. Substitute 2 _ 3 Subtract 2 _ 3 CE for CG. CE from both sides. Substitute 2.4 for GE. 7.2 = CE Multiply both sides by 3. In △JKL, ZW = 7, and LX = 8.1. Find each length. 1a. KW 1b. LZ E X

A M P L E 2 Problem-Solving Application The diagram shows the plan for a triangular piece of a mobile. Where should the sculptor attach the support so that the triangle is balanced? Understand the Problem The answer will be the coordinates of the centroid of △PQR. The important information is the location of the vertices, P (3, 0), Q (0, 8), and R (6, 4). Make a Plan The centroid of the triangle is the point of intersection of the three medians. So write the equations for two medians and find their point of intersection. Solve Let M be the midpoint of ̶̶ QR and N be the midpoint of ̶̶ QP. 3, 61.5, 4) ̶̶̶ PM is vertical. Its equation is x = 3. Its equation is y = 4. The coordinates of the centroid are S (3, 4). ̶̶ RN is horizontal. 5- 3 Medians and Altitudes of Triangles 315 315 ��123 Look Back Let L be the midpoint of intersects x = 3 at S (3, 4). ̶̶ PR. The equation for ̶̶ QL is y = - 4 _ 3 x + 8, which The height of a triangle is the length of an altitude. 2. Find the average of the x-coordinates and the average of the y-coordinates of the vertices of △PQR. Make a conjecture about the centroid of a triangle. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. ̶̶ QY is inside the triangle, but ̶̶ SZ are not. Notice that the lines containing In △QRS, altitude and the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle. ̶̶ RX E X A M P L E 3 Finding the Orthocenter Find the orthocenter of △JKL with vertices J (-4, 2), K (-2, 6), and L (2, 2). Step 1 Graph the triangle. Step 2 Find an equation of the line containing the altitude from K to ̶̶ JL. JL is horizontal, the altitude is

Since   vertical. The line containing it must pass through K (-2, 6), so the equation of the line is x = -2. Step 3 Find an equation of the line containing the altitude from J to ̶̶ KL. slope of   KL = 2 - 6 _ = -1 2 - (-2) The slope of a line perpendicular to   KL is 1. This line must pass through J (-4, 2). y - y 1 = m ( - (-4 Point-slope form Substitute 2 for y 1, 1 for m, and -4 for x 1. Distribute 1. Add 2 to both sides. Step 4 Solve the system to find the coordinates of the orthocenter. ⎧ x = -2 + 6 = 4 Substitute -2 for x. The coordinates of the orthocenter are (-2, 4). 3. Show that the altitude to ̶̶ JK passes through the orthocenter of △JKL. 316 316 Chapter 5 Properties and Attributes of Triangles 4�� THINK AND DISCUSS 1. Draw a triangle in which a median and an altitude are the same segment. What type of triangle is it? 2. Draw a triangle in which an altitude is also a side of the triangle. What type of triangle is it? 3. The centroid of a triangle divides each median into two segments. What is the ratio of the two lengths of each median? 4. GET ORGANIZED Copy and complete the graphic organizer. Fill in the blanks to make each statement true. 5-3 Exercises Exercises KEYWORD: MG7 5-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question.? of a triangle is located 2 __ 3 of the distance from each vertex to the ̶̶̶̶ 1. The midpoint of the opposite side. (centroid or orthocenter) 2. The? of a triangle is perpendicular to the line containing a side. ̶̶̶̶ (altitude or median VX = 204, and RW = 104. Find each length. p. 315 3. VW 5. RY 4. WX 6. WY. Design The diagram shows a plan for p. 315 a

piece of a mobile. A chain will hang from the centroid of the triangle. At what coordinates should the artist attach the chain Multi-Step Find the orthocenter of a triangle with the given vertices. p. 316 8. K (2, -2), L (4, 6), M (8, -2) 9. U (-4, -9), V (-4, 6), W (5, -3) 10. P (-5, 8), Q (4, 5), R (-2, 5) 11. C (-1, -3), D (-1, 2), E (9, 2) 5- 3 Medians and Altitudes of Triangles 317 317 �� Independent Practice For See Exercises Example 12–15 16 17–20 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING PA = 2.9, and HC = 10.8. Find each length. 12. PC 14. JA 13. HP 15. JP 16. Design In the plan for a table, the triangular top has coordinates (0, 10), (4, 0), and (8, 14). The tabletop will rest on a single support placed beneath it. Where should the support be attached so that the table is balanced? Multi-Step Find the orthocenter of a triangle with the given vertices. 17. X (-2, -2), Y (6, 10), Z (6, -6) 18. G (-2, 5), H (6, 5), J (4, -1) 19. R (-8, 9), S (-2, 9), T (-2, 1) 20. A (4, -3), B (8, 5), C (8, -8) Find each measure. 21. GL 23. HL 22. PL 24. GJ 25. perimeter of △GHJ 26. area of △GHJ Algebra Find the centroid of a triangle with the given vertices. 27. A (0, -4), B

(14, 6), C (16, -8) 28. X (8, -1), Y (2, 7), Z (5, -3) Find each length. 29. PZ 31. QZ 30. PX 32. YZ Math History 33. Critical Thinking Draw an isosceles triangle and its line of symmetry. What are four other names for this segment? Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 34. A median of a triangle bisects one of the angles. 35. If one altitude of a triangle is in the triangle’s exterior, then a second altitude is also in the triangle’s exterior. 36. The centroid of a triangle lies in its exterior. 37. In an isosceles triangle, the altitude and median from the vertex angle are the same line as the bisector of the vertex angle. 38. Write a two-column proof. ̶̶ PS and Given: Prove: △PQR is an isosceles triangle. ̶̶ RT are medians of △PQR. ̶̶ PS ≅ ̶̶ RT In 1678, Giovanni Ceva published his famous theorem that states the conditions necessary for three Cevians (segments from a vertex of a triangle to the opposite side) to be concurrent. The medians and altitudes of a triangle meet these conditions. Plan: Show that △PTR ≅ △RSP and use CPCTC to conclude that ∠QPR ≅ ∠QRP. 39. Write About It Draw a large triangle on a sheet of paper and cut it out. Find the centroid by paper folding. Try to balance the shape on the tip of your pencil at a point other than the centroid. Now try to balance the shape at its centroid. Explain why the centroid is also called the center of gravity. 318 318 Chapter 5 Properties and Attributes of Triangles �� 40. This problem will prepare you for the Multi-Step TAKS Prep on page 328. The towns of Davis, El Monte, and Fairview have the coordinates shown in the table, where each unit of the coordinate plane represents one mile. A music company has stores in each city and a distribution warehouse at the centroid of △DEF. a. What are the coordinates of the warehouse? b.

Find the distance from the warehouse to the City Location Davis El Monte Fairview D (0, 0) E (0, 8) F (8, 0) Davis store. Round your answer to the nearest tenth of a mile. c. A straight road connects El Monte and Fairview. What is the distance from the warehouse to the road? 41. ̶̶ RV, and ̶̶ QT, NOT necessarily true? ̶̶̶ SW are medians of △QRS. Which statement is QP = 2 _ 3 RP = 2PV QT RT = ST QT = SW 42. Suppose that the orthocenter of a triangle lies outside the triangle. Which points of concurrency are inside the triangle? I. incenter II. circumcenter III. centroid I and II only I and III only II and III only I, II, and III 43. In the diagram, which of the following correctly describes ̶̶ LN? Altitude Median Angle bisector Perpendicular bisector CHALLENGE AND EXTEND 44. Draw an equilateral triangle. a. Explain why the perpendicular bisector of any side contains the vertex opposite that side. b. Explain why the perpendicular bisector through any vertex also contains the median, the altitude, and the angle bisector through that vertex. c. Explain why the incenter, circumcenter, centroid, and orthocenter are the same point. 45. Use coordinates to show that the lines containing ̶̶ RS, ̶̶ ST, and the altitudes of a triangle are concurrent. a. Find the slopes of b. Find the slopes of lines ℓ, m, and n. c. Write equations for lines ℓ, m, and n. d. Solve a system of equations to find the point P where lines ℓ and m intersect. ̶̶ RT. e. Show that line n contains P. f. What conclusion can you draw? 5- 3 Medians and Altitudes of Triangles 319 319 �� SPIRAL REVIEW 46. At a baseball game, a bag of peanuts costs $0.75 more than a bag of popcorn. If a family purchases 5 bags of peanuts and 3 bags of popcorn for $21.75, how much does one bag of peanuts cost? (Previous course) Determine if each biconditional is true. If false, give a counterexample. (Less

on 2-4) 47. The area of a rectangle is 40 cm 2 if and only if the length of the rectangle is 4 cm and the width of the rectangle is 10 cm. 48. A nonzero number n is positive if and only if -n is negative. ̶̶ QP, and ̶̶ NQ, Find each measure. (Lesson 5-2) ̶̶̶ QM are perpendicular bisectors of △JKL. 49. KL 50. QJ 51. m∠JQL Construction Orthocenter of a Triangle    Draw a large scalene acute triangle ABC on a piece of patty paper. Find the altitude of each side by folding the side so that it overlaps itself and so that the fold intersects the opposite vertex. Mark the point where the three lines containing the altitudes intersect and label it P. P is the orthocenter of △ABC. 1. Repeat the construction for a scalene obtuse triangle 2. Make a conjecture about the location of the and a scalene right triangle. orthocenter in an acute, an obtuse, and a right triangle. Q: What high school math classes did you take? A: Algebra 1, Geometry, and Statistics. KEYWORD: MG7 Career Q: What type of training did you receive? A: In high school, I took classes in electricity, electronics, and drafting. I began an apprenticeship program last year to prepare for the exam to get my license. Q: How do you use math? A: Determining the locations of outlets and circuits on blueprints requires good spatial sense. I also use ratios and proportions, calculate distances, work with formulas, and estimate job costs. Alex Peralta Electrician 320 320 Chapter 5 Properties and Attributes of Triangles �� 5-3 Special Points in Triangles In this lab you will use geometry software to explore properties of the four points of concurrency you have studied. Use with Lesson 5-3 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.B KEYWORD: MG7 Lab5 Activity 1 Construct a triangle. 2 Construct the perpendicular bisector of each side of the triangle. Construct the point of intersection of these three lines. This is the circumcenter of the triangle. Label it U and hide the

perpendicular bisectors. 3 4 5 In the same triangle, construct the bisector of each angle. Construct the point of intersection of these three lines. This is the incenter of the triangle. Label it I and hide the angle bisectors. In the same triangle, construct the midpoint of each side. Then construct the three medians. Construct the point of intersection of these three lines. Label the centroid C and hide the medians. In the same triangle, construct the altitude to each side. Construct the point of intersection of these three lines. Label the orthocenter O and hide the altitudes. 6 Move a vertex of the triangle and observe the positions of the four points of concurrency. In 1765, Swiss mathematician Leonhard Euler showed that three of these points are always collinear. The line containing them is called the Euler line. Try This 1. Which three points of concurrency lie on the Euler line? 2. Make a Conjecture Which point on the Euler line is always between the other two? Measure the distances between the points. Make a conjecture about the relationship of the distances between these three points. 3. Make a Conjecture Move a vertex of the triangle until all four points of concurrency are collinear. In what type of triangle are all four points of concurrency on the Euler line? 4. Make a Conjecture Find a triangle in which all four points of concurrency coincide. What type of triangle has this special property? 5- 3 Technology Lab 321 321 5-4 The Triangle Midsegment Theorem TEKS G.7.B Dimensionality and the geometry of location: use slopes and equations of lines to investigate... special segments of triangles and other polygons. Objective Prove and use properties of triangle midsegments. Vocabulary midsegment of a triangle Why learn this? You can use triangle midsegments to make indirect measurements of distances, such as the distance across a volcano. (See Example 3.) Also G.2.A, G.2.B, G.3.B, G.5.A, G.9.B A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. E X A M P L E 1 Examining Midsegments in the Coordinate Plane In △GHJ, show that midsegment parallel to GJ

and that KL = 1 __ ̶̶ Step 1 Find the coordinates of K and L. GJ. 2 ̶̶ KL is mdpt. of ̶̶̶ GH = ( ) -7 + (-5) -2 + 6 _ _, 2 2 = (-6, 2) mdpt. of ̶̶ HJ = ( 6 + 2 -5 + 1 _ _, 2 2 ) = (-2, 4) Step 2 Compare the slopes of slope of ̶̶ KL = ̶̶ GJ. ̶̶ KL and = 1 _ 2 4 - 2 _ -2 - (-6) slope of ̶̶ GJ = 2 - (-2) _ 1 - (-7) = 1 _ 2 Since the slopes are the same, ̶̶ KL and Step 3 Compare the lengths of ̶̶ KL ǁ ̶̶ GJ. ̶̶ GJ. 2 + (4 - 2) 2 = 2 √  5 KL = √  ⎤ ⎦ ⎡ ⎣ -2 - (-6) GJ = √  ⎤ ⎦ ⎤ ⎦ ⎡ ⎣ ⎡ ⎣ 2 - (-2) 1 - (-7) + (4 √  5 ), KL = 1 _ Since 2 √  5 = 1 _ 2 2 GJ. 2 2 = 4 √  5 1. The vertices of △RST are R (-7, 0), S (-3, 6), and T (9, 2). ̶̶ RT, and N is the midpoint of ̶̶ ST. M is the midpoint of Show that ̶̶̶ MN ǁ ̶̶ RS and MN = 1 __ 2 RS. 322 322 Chapter 5 Properties and Attributes of Triangles �� The relationship shown in Example 1 is true for the three midsegments of every triangle. Theorem 5-4-1 Triangle Midsegment Theorem

A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. ̶̶ DE ǁ ̶̶ AC, DE = 1_ 2 AC You will prove Theorem 5-4-1 in Exercise 38. E X A M P L E 2 Using the Triangle Midsegment Theorem Find each measure. A UW ST UW = 1 _ 2 UW = 1 _ (7.4) 2 UW = 3.7 △ Midsegment Thm. Substitute 7.4 for ST. Simplify. B m∠SVU ̶̶ ̶̶̶ UW ǁ ST m∠SVU = m∠VUW m∠SVU = 41° △ Midsegment Thm. Alt. Int.  Thm. Substitute 41° for m∠VUW. Find each measure. 2a. JL 2b. PM 2c. m∠MLK E X A M P L E 3 Indirect Measurement Application Anna wants to find the distance across the base of Capulin Volcano, an extinct volcano in New Mexico. She measures a triangle at one side of the volcano as shown in the diagram. What is AE? BD = 1 _ 2 775 = 1 _ 2 AE AE △ Midsegment Thm. Substitute 775 for BD. 1550 = AE Multiply both sides by 2. The distance AE across the base of the volcano is about 1550 meters. 3. What if…? Suppose Anna’s result in Example 3 is correct. To check it, she measures a second triangle. How many meters will she measure between H and F? 323 5- 4 The Triangle Midsegment Theorem 323 640 m1005 m640 m1005 mge07se_c05l04007aGFHAE��700 m920 m920 m775 m700 mge07se_c05l04006aCDBAE THINK AND DISCUSS 1. Explain why ̶̶ XY is NOT a midsegment of the triangle. 2. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of a triangle midsegment and list its properties. Then draw an example and a nonexample. 5-4 Exercises Exercises KEYWORD: MG7 5-4 KEYWORD:

MG7 Parent GUIDED PRACTICE 1. Vocabulary The midsegment of a triangle joins the triangle. (endpoints or midpoints)? of two sides of the ̶̶̶̶. 322 2. The vertices of △PQR are P (-4, -1), Q (2, 9), and R (6, 3). S is the midpoint of and T is the midpoint of ̶̶ QR. Show that ̶̶ ST ǁ ̶̶ PR and ST = 1 __ 2 PR. ̶̶ PQ, Find each measure. p. 323 3. NM 5. NZ 7. m∠YXZ 4. XZ 6. m∠LMN 8. m∠XLM. 323 Independent Practice For See Exercises Example 10 11–16 17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 9. Architecture In this A-frame house, ̶̶ XZ is 30 feet. ̶̶ CD is slightly above the width of the first floor The second floor and parallel to the midsegment of △XYZ. Is the width of the second floor more or less than 5 yards? Explain. � � � � � PRACTICE AND PROBLEM SOLVING 10. The vertices of △ABC are A (-6, 11), B (6, -3), and C (-2, -5). D is the midpoint ̶̶ AB. Show that ̶̶ AC, and E is the midpoint of ̶̶ DE ǁ ̶̶ CB and DE = 1 __ 2 CB. of Find each measure. 11. GJ 13. RJ 12. RQ 14. m∠PQR 15. m∠HGJ 16. m∠GPQ 324 324 Chapter 5 Properties and Attributes of Triangles �� 17. Carpentry In each support for the garden swing, ̶̶ BA and ̶̶ DE is attached at the midpoints ̶̶ BC. The distance AC is 4 1 __ 2 feet. the crossbar of legs The carpenter has a timber that is 30 inches long. Is this timber long enough to be used as one of the crossbars? Explain. △K

LM is the midsegment triangle of △GHJ. 18. What is the perimeter of △GHJ? 19. What is the perimeter of △KLM? 20. What is the relationship between the perimeter of △GHJ and the perimeter of △KLM? Algebra Find the value of n in each triangle. 21. 24. 22. 25. 23. 26. 27. /////ERROR ANALYSIS///// Below are two solutions for finding BC. Which is incorrect? Explain the error. 28. Critical Thinking Draw scalene △DEF. Label X as the midpoint of Y as the midpoint of midpoints. List all of the congruent angles in your drawing. ̶̶ EF, and Z as the midpoint of ̶̶ DE, ̶̶ DF. Connect the three 29. Estimation The diagram shows the sketch for a new street. Parallel parking spaces will be painted on both sides of the street. Each parallel parking space is 23 feet long. About how many parking spaces can the city accommodate on both sides of the new street? Explain your answer. ̶̶ EH, and ̶̶ CG, and △GHE, respectively. Find each measure. ̶̶ FJ are midsegments of △ABD, △GCD, 30. CG 31. EH 32. FJ 33. m∠DCG 34. m∠GHE 35. m∠FJH 36. Write About It An isosceles triangle has two congruent sides. Does it also have two congruent midsegments? Explain. 325 5- 4 The Triangle Midsegment Theorem 325 ��ge07sec05l04004aABoehmMarket Street (440 ft)Springfield RoadLake AvenueNew street�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 328. The figure shows the roads connecting towns A, B, and C. A music company has a store in each town and a distribution warehouse W at the midpoint of road a. What is the distance from the warehouse to point X? b. A truck starts at the warehouse, delivers instruments to the stores in towns A, B, and C (

in this order) and then returns to the warehouse. What is the total length of the trip, assuming the driver takes the shortest possible route? ̶̶ XY. 38. Use coordinates to prove the Triangle Midsegment Theorem. a. M is the midpoint of b. N is the midpoint of ̶̶ c. Find the slopes of PR and ̶̶ PQ. What are its coordinates? ̶̶ QR. What are its coordinates? ̶̶̶ MN. What can you conclude? d. Find PR and MN. What can you conclude? 39. ̶̶ PQ is a midsegment of △RST. What is the length of ̶̶ RT? 9 meters 21 meters 45 meters 63 meters 40. In △UVW, M is the midpoint of ̶̶̶ VW. Which statement is true? midpoint of ̶̶ VU, and N is the VM = VN MN = UV VU = 2VM VW = 1 _ 2 VN 41. △XYZ is the midsegment triangle of △JKL, XY = 8, YK = 14, and m∠YKZ = 67°. Which of the following measures CANNOT be determined? KL JY m∠XZL m∠KZY CHALLENGE AND EXTEND 42. Multi-Step The midpoints of the sides of a triangle are A (-6, 3), B (2, 1), and C (0, -3). Find the coordinates of the vertices of the triangle. 43. Critical Thinking Classify the midsegment triangle of an equilateral triangle by its side lengths and angle measures. Algebra Find the value of n in each triangle. 44. 45. 326 326 Chapter 5 Properties and Attributes of Triangles �� 46. △XYZ is the midsegment triangle of △PQR. Write a congruence statement involving all four of the smaller triangles. What is the relationship between the area of △XYZ and △PQR? ̶̶ AB is a midsegment of △XYZ. is a midsegment of △CDZ, and a. Copy and complete the table. 47. ̶̶ CD is a midsegment of △ABZ

. ̶̶̶ GH is a midsegment of △EFZ. ̶̶ EF 1 2 3 4 Number of Midsegment Length of Midsegment b. If this pattern continues, what will be the length of midsegment 8? c. Write an algebraic expression to represent the length of midsegment n. (Hint: Think of the midsegment lengths as powers of 2.) SPIRAL REVIEW Suppose a 2% acid solution is mixed with a 3% acid solution. Find the percent of acid in each mixture. (Previous course) 48. a mixture that contains an equal amount of 2% acid solution and 3% acid solution 49. a mixture that contains 3 times more 2% acid solution than 3% acid solution A figure has vertices G (-3, -2), H (0, 0), J (4, 1), and K (1, -2). Given the coordinates of the image of G under a translation, find the coordinates of the images of H, J, and K. (Lesson 1-7) 50. (-3, 2) 51. (1, -4) 52. (3, 0) Find each length. (Lesson 5-3) 53. NX 54. MR 55. NP Construction Midsegment of a Triangle    Draw a large triangle. Label the vertices A, B, and C. ̶̶ AB ̶̶ BC. Label the midpoints X Construct the midpoints of and and Y, respectively. Draw the midsegment ̶̶ XY. 1. Using a ruler, measure ̶̶ XY and ̶̶ AC. How are the two lengths related? 2. How can you use a protractor to verify that ̶̶ XY is parallel to ̶̶ AC? 327 5- 4 The Triangle Midsegment Theorem 327 �� SECTION 5A Segments in Triangles Location Contemplation A chain of music stores has locations in Ashville, Benton, and Carson. The directors of the company are using a coordinate plane to decide on the location for a new distribution warehouse. Each unit on the plane represents one mile. 1. A plot of land is available at the centroid of the triangle formed by the three cities. What are the coordinates for this location? 2. If the directors build the warehouse at the centroid, about how far will

it be from each of the cities? 3. Another plot of land is available at the orthocenter of the triangle. What are the coordinates for this location? 4. About how far would the warehouse be from each city if it were built at the orthocenter? 5. A third option is to build the warehouse at the circumcenter of the triangle. What are the coordinates for this location? 6. About how far would the warehouse be from each city if it were built at the circumcenter? 7. The directors decide that the warehouse should be equidistant from each city. Which location should they choose? 328 328 Chapter 5 Properties and Attributes of Triangles �� SECTION 5A Quiz for Lessons 5-1 Through 5-4 5-1 Perpendicular and Angle Bisectors Find each measure. 1. PQ 2. JM 3. AC 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints M (-1, -3) and N (7, 1). 5-2 Bisectors of Triangles 5. ̶̶ PY, and ̶̶ PX, bisectors of △RST. Find PS and XT. ̶̶ PZ are the perpendicular 6. ̶̶ JK and ̶̶ HK are angle bisectors of △GHJ. Find m∠GJK and the distance from K to ̶̶ HJ. 7. Find the circumcenter of △TVO with vertices T (9, 0), V (0, -4), and O (0, 0). 5-3 Medians and Altitudes of Triangles 8. In △DEF, BD = 87, and WE = 38. Find BW, CW, and CE. 9. Paula cuts a triangle with vertices at coordinates (0, 4), (8, 0), and (10, 8) from grid paper. At what coordinates should she place the tip of a pencil to balance the triangle? 10. Find the orthocenter of △PSV with vertices P (2, 4), S (8, 4), and V (4, 0). 5-4 The Triangle Midsegment Theorem 11. Find ZV, PM, and m∠RZV in △JMP. 12. What is the distance XZ across

the pond? � �� Ready to Go On? 329 329 �� Solving Compound Inequalities Algebra To solve an inequality, you use the Properties of Inequality and inverse operations to undo the operations in the inequality one at a time. See Skills Bank page S60 Properties of Inequality PROPERTY ALGEBRA Addition Property Subtraction Property Multiplication Property Division Property Transitive Property Comparison Property If a < b, then a + c 0, then ac < bc. If a bc. If a 0, then a _ c b _ c. If a 0, then a < c. A compound inequality is formed when two simple inequalities are combined into one statement with the word and or or. To solve a compound inequality, solve each simple inequality and find the intersection or union of the solutions. The graph of a compound inequality may represent a line, a ray, two rays, or a segment. Example Solve the compound inequality 5 < 20 - 3a ≤ 11. What geometric figure does the graph represent? 5 < 20 - 3a AND 20 - 3a ≤ 11 Rewrite the compound inequality as two -15 < -3a 5 > a AND AND -3a ≤ -9 Subtract 20 from both sides. simple inequalities. a ≥ 3 Divide both sides by -3 and reverse the inequality symbols. 3 ≤ a < 5 Combine the two solutions into a single statement. The graph represents a segment. Try This TAKS Grades 9–11 Obj. 4 Solve. What geometric figure does each graph represent? 1. -4 + x > 1 OR -8 + 2x < -6 2. 2x - 3 ≥ -5 OR x - 4 > -1 3. -6 < 7 - x ≤ 12 5. 3x ≥ 0 OR x + 5 < 7 4. 22 < -2 - 2x ≤ 54 6. 2x - 3 ≤ 5 OR -2x + 3 ≤

-9 330 330 Chapter 5 Properties and Attributes of Triangles �� 5-5 Use with Lesson 5-5 Activity 1 Explore Triangle Inequalities Many of the triangle relationships you have learned so far involve a statement of equality. For example, the circumcenter of a triangle is equidistant from the vertices of the triangle, and the incenter is equidistant from the sides of the triangle. Now you will investigate some triangle relationships that involve inequalities. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about... polygons and their component parts based on explorations and concrete models. Also G.5.B 1 Draw a large scalene triangle. Label the vertices A, B, and C. 2 Measure the sides and the angles. Copy the table below and record the measures in the first row. BC AC AB m∠A m∠B m∠C Triangle 1 Triangle 2 Triangle 3 Triangle 4 Try This 1. In the table, draw a circle around the longest side length, and draw a circle around the greatest angle measure of △ABC. Draw a square around the shortest side length, and draw a square around the least angle measure. 2. Make a Conjecture Where is the longest side in relation to the largest angle? Where is the shortest side in relation to the smallest angle? 3. Draw three more scalene triangles and record the measures in the table. Does your conjecture hold? Activity 2 1 Cut three sets of chenille stems to the following lengths. 3 inches, 4 inches, 6 inches 3 inches, 4 inches, 7 inches 3 inches, 4 inches, 8 inches 2 Try to make a triangle with each set of chenille stems. Try This 4. Which sets of chenille stems make a triangle? 5. Make a Conjecture For each set of chenille stems, compare the sum of any two lengths with the third length. What is the relationship? 6. Select a different set of three lengths and test your conjecture. Does your conjecture hold? 5- 5 Geometry Lab 331 331 5-5 Indirect Proof and Inequalities in One Triangle TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.C, G.3.E, G.5.B Objectives Write indirect proofs. Apply inequalities in one triangle. Why learn this? You can use a triangle inequality to find a reasonable range

of values for an unknown distance. (See Example 5.) Vocabulary indirect proof So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Writing an Indirect Proof 1. Identify the conjecture to be proven. 2. Assume the opposite (the negation) of the conclusion is true. 3. Use direct reasoning to show that the assumption leads to a contradiction. 4. Conclude that since the assumption is false, the original conjecture must be true. E X A M P L E 1 Writing an Indirect Proof Write an indirect proof that a right triangle cannot have an obtuse angle. Step 1 Identify the conjecture to be proven. Given: △JKL is a right triangle. Prove: △JKL does not have an obtuse angle. Step 2 Assume the opposite of the conclusion. Assume △JKL has an obtuse angle. Let ∠K be obtuse. Step 3 Use direct reasoning to lead to a contradiction. m∠K + m∠L = 90° The acute  of a rt. △ are comp. m∠K = 90° - m∠L m∠K > 90° Subtr. Prop. of = Def. of obtuse ∠ 90° - m∠L > 90° m∠L < 0° Substitute 90° - m∠L for m∠K. Subtract 90° from both sides and solve for m∠L. However, by the Protractor Postulate, a triangle cannot have an angle with a measure less than 0°. Step 4 Conclude that the original conjecture is true. The assumption that △JKL has an obtuse angle is false. Therefore △JKL does not have an obtuse angle. 1. Write an indirect proof that a triangle cannot have two right angles. 332 332 Chapter 5 Properties and Attributes of Triangles �� The positions of the longest and shortest sides of a triangle are related to the positions of the largest

and smallest angles. Theorems Angle-Side Relationships in Triangles THEOREM HYPOTHESIS CONCLUSION 5-5-1 If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. (In △, larger ∠ is opp. longer side.) 5-5-2 If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (In △, longer side is opp. larger ∠.) m∠C > m∠A XY > XZ AB > BC m∠Z > m∠Y You will prove Theorem 5-5-1 in Exercise 67. PROOF PROOF Theorem 5-5-2 Given: m∠P > m∠R Prove: QR > QP Consider all cases when you assume the opposite. If the conclusion is QR > QP, the negation includes QR < QP and QR = QP. Indirect Proof: Assume QR ≯ QP. This means that either QR < QP or QR = QP. Case 1 the longer side. This contradicts the given information. So QR ≮ QP. If QR < QP, then m∠P < m∠R because the larger angle is opposite Case 2 This also contradicts the given information, so QR ≠ QP. If QR = QP, then m∠P = m∠R by the Isosceles Triangle Theorem. The assumption QR ≯ QP is false. Therefore QR > QP. E X A M P L E 2 Ordering Triangle Side Lengths and Angle Measures A Write the angles in order from smallest to largest. ̶̶ GJ, so the smallest angle is ∠H. ̶̶ HJ, so the largest angle is ∠G. The shortest side is The longest side is The angles from smallest to largest are ∠H, ∠J, and ∠G. B Write the sides in order from shortest to longest. m∠M = 180° - (39° + 54°) = 87° △ Sum Thm. ̶̶̶ The smallest angle is ∠L, so the shortest side is KM. ̶̶ KL. The largest angle is ∠M, so the longest side is ̶̶̶ LM, and The sides from shortest to longest are ̶̶̶

KM, ̶̶ KL. 2a. Write the angles in order from smallest to largest. 2b. Write the sides in order from shortest to longest. 5- 5 Indirect Proof and Inequalities in One Triangle 333 333 �� A triangle is formed by three segments, but not every set of three segments can form a triangle. Segments with lengths of 7, 4, and 4 can form a triangle. Segments with lengths of 7, 3, and 3 cannot form a triangle. A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Theorem 5-5-3 Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. AB + BC > AC BC + AC > AB AC + AB > BC You will prove Theorem 5-5-3 in Exercise 68. E X A M P L E 3 Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. A 3, 5 ✓ 10 > 5 ✓ 12 > 3 ✓ Yes—the sum of each pair of lengths is greater than the third length. B 4, 6.5, 11 4 + 6.5  11 10.5 ≯ 11 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. C n + 5, n 2, 2n, when n = 3 Step 1 Evaluate each expression when n = 3 2n 2 (3) 6 To show that three lengths cannot be the side lengths of a triangle, you only need to show that one of the three triangle inequalities is false. Step 2 Compare the lengths 17 > 6 ✓ 14 > 9 ✓ 15 > 8 ✓ Yes—the sum of each pair of lengths is greater than the third length. Tell whether a triangle can have sides with the given lengths. Explain. 3a. 8, 13, 21 3c. t - 2, 4t, t 2 + 1, when t = 4 3b. 6.2, 7, 9 334 334 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 4 Finding Side Lengths The lengths of two sides of a triangle are 6 centimeters and 11 centimeters. Find the range of possible lengths for the third side. Let s represent the length of the third side. Then apply the Triangle Inequality Theorem. s + 6 > 11 s

+ 11 > 6 s > 5 s > -5 6 + 11 > s 17 > s Combine the inequalities. So 5 < s < 17. The length of the third side is greater than 5 centimeters and less than 17 centimeters. 4. The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side. E X A M P L E 5 Travel Application The map shows the approximate distances from San Antonio to Mason and from San Antonio to Austin. What is the range of distances from Mason to Austin? Let d be the distance from Mason to Austin. d + 111 > 78 d + 78 > 111 111 + 78 > d △ Inequal. Thm. d > -33 d > 33 189 > d Subtr. Prop. of Inequal. 33 < d < 189 Combine the inequalities. The distance from Mason to Austin is greater than 33 miles and less than 189 miles. 5. The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City? THINK AND DISCUSS 1. To write an indirect proof that an angle is obtuse, a student assumes that the angle is acute. Is this the correct assumption? Explain. 2. Give an example of three measures that can be the lengths of the sides of a triangle. Give an example of three lengths that cannot be the sides of a triangle. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, explain what you know about △ABC as a result of the theorem. 5- 5 Indirect Proof and Inequalities in One Triangle 335 335 AustinSanMarcosJohnsonCitySeguinSan AntonioMason10103535290281377183879078 mi111 mige07se_c05l05002aABeckmann�� 5-5 Exercises Exercises KEYWORD: MG7 5-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe the process of an indirect proof in your own words Write an indirect proof of each statement. p. 332 2. A scalene triangle cannot have two congruent angles. 3. An isosceles triangle cannot have a base angle that is a right angle. Write the angles in order p. 333 from smallest

to largest. 5. Write the sides in order from shortest to longest Tell whether a triangle can have sides with the given lengths. Explain. p. 334 6. 4, 7, 10 7. 2, 9, 12, 3 1 _ 8. 3 1 _ 2 2, 6 9. 3, 1.1, 1.7 10. 3x, 2x - 1, x 2, when x = 5 11. 7c + 6, 10c - 7, 3 c 2, when. 335 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. 12. 8 mm, 12 mm 13. 16 ft, 16 ft 14. 11.4 cm, 12 cm. 335 15. Design The refrigerator, stove, and sink in a kitchen are at the vertices of a path called the work triangle. a. If the angle at the sink is the largest, which side of the work triangle will be the longest? b. The designer wants the longest side of this triangle to be 9 feet long. Can the lengths of the other sides be 5 feet and 4 feet? Explain. �� Independent Practice Write an indirect proof of each statement. PRACTICE AND PROBLEM SOLVING 16. A scalene triangle cannot have two congruent midsegments. 17. Two supplementary angles cannot both be obtuse angles. 18. Write the angles in order from smallest to largest. �� 19. Write the sides in order �� from shortest to longest. For See Exercises Example 16–17 18–19 20–25 26–31 32 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 Tell whether a triangle can have sides with the given lengths. Explain. 20. 6, 10, 15 21. 14, 18, 32 22. 11.9, 5.8, 5.8 23. 103, 41.9, 62.5 24. z + 8, 3z + 5, 4z - 11, when z = 6 25. m + 11, 8m, m 2 + 1, when m = 3 336 336 Chapter 5 Properties and Attributes of Triangles �� The lengths of two sides of a triangle are given. Find the range of possible lengths The lengths of two sides of a triangle are given. Find the range of possible lengths for the

third side. for the third side. Bicycles 26. 26. 4 yd, 19 yd 29. 3.07 m, 1.89 m 27. 28 km, 23 km in., 3 5_ 30. 2 1_ 8 8 in. 28. 9.2 cm, 3.8 cm 31. 3 5_ ft, 6 1_ 2 6 ft 32. Bicycles The five steel tubes of this mountain bike frame form two triangles. List the five tubes in order from shortest to longest. Explain your answer. 33. Critical Thinking The length of the base of an isosceles triangle is 15. What is the range of possible lengths for each leg? Explain. List the sides of each triangle in order from shortest to longest. 34. 35. Lance Armstrong, of Austin, broke his own record in 2005 and became the first person to win seven consecutive titles in the Tour de France cycling competition. Only one other person has won five consecutive times. In each set of statements, name the two that contradict each other. 36. △PQR is a right triangle. 37. ∠Y is supplementary to ∠Z. △PQR is a scalene triangle. △PQR is an acute triangle. 38. △JKL is isosceles with base ̶̶ JL. In △JKL, m∠K > m∠J In △JKL, JK > LK 40. Figure A is a polygon. Figure A is a triangle. Figure A is a quadrilateral. m∠Y < 90° ∠Y is an obtuse angle. 39. ̶̶ AB ⊥ ̶̶ AB ≅ ̶̶ AB ǁ ̶̶ BC ̶̶ CD ̶̶ BC 41. x is even. x is a multiple of 4. x is prime. Compare. Write <, >, or =. PS 42. QS 44. QS 46. PQ QR RS 43. PQ QS 45. QS 47. RS RS PS 48. m∠ABE m∠BEA 49. m∠CBE m∠CEB 50. m∠DCE m∠DEC 51. m∠DCE m∠CDE 52. m∠ABE m∠EAB 53. m∠EBC m∠ECB List the

angles of △JKL in order from smallest to largest. 54. J (-3, -2), K (3, 6), L (8, -2) 55. J (-5, -10), K (-5, 2), L (7, -5) 56. J (-4, 1), K (-3, 8), L (3, 4) 57. J (-10, -4), K (0, 3), L (2, -8) 58. Critical Thinking An attorney argues that her client did not commit a burglary because a witness saw her client in a different city at the time of the burglary. Explain how this situation is an example of indirect reasoning. 5- 5 Indirect Proof and Inequalities in One Triangle 337 337 B54.1 cm50.8 cmADCge07se_ c05105005aa1st Pass2/7/05N Patel64º66º56º50º�� 59. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes between four cities. a. The airline’s planes fly at an average speed of 500 mi/h. What is the range of time it might take to fly from Auburn (A) to Raymond (R)? b. The airline offers one frequent-flier mile for every mile flown. Is it possible to earn 1800 miles by flying from Millford (M) to Auburn (A)? Explain. Multi-Step Each set of expressions represents the lengths of the sides of a triangle. Find the range of possible values of n. 60. n, 6, 8 61. 2n, 5, 7 62. n + 1, 3, 6 63. n + 1, n + 2, n + 3 64. n + 2, n + 3, 3n - 2 65. n, n + 2, 2n + 1 66. Given that P is in the interior of △XYZ, prove that XY + XP + PZ > YZ. 67. Complete the proof of Theorem 5-5-1 by filling in the blanks. Given: RS > RQ Prove: m∠RQS > m∠S Proof: ̶̶ RS so that RP = RQ. So?, and m∠

1 = m∠2 by c. ̶̶̶̶ Locate P on by b. m∠RQS = d. Inequality. Then m∠RQS > m∠2 by e. m∠2 = m∠3 + f. Inequality. Therefore m∠RQS > m∠S by g.?. So m∠RQS > m∠1 by the Comparison Property of ̶̶̶̶?. So m∠2 > m∠S by the Comparison Property of ̶̶̶̶?. By the Exterior Angle Theorem, ̶̶̶̶ ̶̶ RQ by a. ̶̶ RP ≅?. By the Angle Addition Postulate, ̶̶̶̶?. Then ∠1 ≅ ∠2 ̶̶̶̶?. ̶̶̶̶ 68. Complete the proof of the Triangle Inequality Theorem. Given: △ABC Prove: AB + BC > AC, AB + AC > BC, AC + BC > AB Proof: One side of △ABC is as long as or longer than each of the other sides. Let this side be remains to be proved is AC + BC > AB. ̶̶ AB. Then AB + BC > AC, and AB + AC > BC. Therefore what Statements Reasons 1. a.? ̶̶̶̶ 2. Locate D on  AC so that BC = DC. 3. AC + DC = b.? ̶̶̶̶ 4. ∠1 ≅ ∠2 5. m∠1 = m∠2 6. m∠ABD = m∠2 + e.? ̶̶̶̶ 7. m∠ABD > m∠2 8. m∠ABD > m∠1 9. AD > AB 10. AC + DC > AB 11. i.? ̶̶̶̶ 1. Given 2. Ruler Post. 3. Seg. Add. Post. 4. c. 5. d.? ̶̶̶̶? ̶̶̶̶ 6. ∠ Add. Post. 7. Comparison Prop. of Inequal. 8. f. 9. g. 10. h.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ 11. Subst. 69.

Write About It Explain why the hypotenuse is always the longest side of a right triangle. Explain why the diagonal of a square is longer than each side. 338 338 Chapter 5 Properties and Attributes of Triangles �� 70. The lengths of two sides of a triangle are 3 feet and 5 feet. Which could be the length of the third side? 3 feet 8 feet 15 feet 16 feet 71. Which statement about △GHJ is false? GH < GJ m∠H > m∠ J GH + HJ < GJ △GHJ is a scalene triangle. 72. In △RST, m∠S = 92°. Which is the longest side of △RST? ̶̶ RS ̶̶ ST ̶̶ RT Cannot be determined CHALLENGE AND EXTEND 73. Probability A bag contains five sticks. The lengths of the sticks are 1 inch, 3 inches, 5 inches, 7 inches, and 9 inches. Suppose you pick three sticks from the bag at random. What is the probability you can form a triangle with the three sticks? 74. Complete this indirect argument that √  2 is irrational. Assume that a.?. ̶̶̶̶ __ q, where p and q are positive integers that have no common factors.?, and p 2 = c. ̶̶̶̶ p Then √  2 = Thus 2 = b. p is even. Since p 2 is the square of an even number, p 2 is divisible by 4 because d.?, and so q is even. Then p and ̶̶̶̶ q have a common factor of 2, which contradicts the assumption that p and q have no common factors.?. But then q 2 must be even because e. ̶̶̶̶?. This implies that p 2 is even, and thus ̶̶̶̶ 75. Prove that the perpendicular segment from a point to a line is the shortest segment from the point to the line. ̶̶ PX ⊥ ℓ. Y is any point on ℓ other than X. Given: Prove: PY > PX Plan: Show that ∠2 and ∠P are complementary. Use the Comparison Property of Inequality to show that 90° > m∠2. Then show that m∠1 > m∠2 and thus PY > PX.

SPIRAL REVIEW Write the equation of each line in standard form. (Previous course) 76. the line through points (-3, 2) and (-1, -2) 77. the line with slope 2 and x-intercept of -3 Show that the triangles are congruent for the given value of the variable. (Lesson 4-4) 78. △PQR ≅ △TUS, when x = -1 79. △ABC ≅ △EFD, when p = 6 Find the orthocenter of a triangle with the given vertices. (Lesson 5-3) 80. R (0, 5), S (4, 3), T (0, 1) 81. M (0, 0), N (3, 0), P (0, 5) 5- 5 Indirect Proof and Inequalities in One Triangle 339 339 �� 5-6 Inequalities in Two Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.E Objective Apply inequalities in two triangles. Who uses this? Designers of this circular swing ride can use the angle of the swings to determine how high the chairs will be at full speed. (See Example 2.) In this lesson, you will apply inequality relationships between two triangles. Theorems Inequalities in Two Triangles THEOREM HYPOTHESIS CONCLUSION 5-6-1 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. 5-6-2 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. m∠A > m∠D BC > EF m∠ J > m∠M GH > KL You will prove Theorem 5-6-1 in Exercise 35. PROOF PROOF Converse of the Hinge Theorem ̶̶ PR ≅ ̶̶ XZ, QR > YZ ̶̶ PQ ≅ Given: Prove: m∠P > m

∠X ̶̶ XY, Indirect Proof: Assume m∠P ≯ m∠X. So either m∠P < m∠X, or m∠P = m∠X. Case 1 If m∠P < m∠X, then QR < YZ by the Hinge Theorem. This contradicts the given information that QR > YZ. So m∠P ≮ m∠X. Case 2 If m∠P = m∠X, then ∠P ≅ ∠X. So △PQR ≅ △XYZ by SAS. Then ̶̶ YZ by CPCTC, and QR = YZ. This also contradicts ̶̶ QR ≅ the given information. So m∠P ≠ m∠X. The assumption m∠P ≯ m∠X is false. Therefore m∠P > m∠X. 340 340 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Hinge Theorem and Its Converse A Compare m∠PQS and m∠RQS. Compare the side lengths in △PQS and △RQS. QS = QS PQ = RQ PS > RS By the Converse of the Hinge Theorem, m∠PQS > m∠RQS. B Compare KL and MN. Compare the sides and angles in △KLN and △MNL. KN = ML LN = LN m∠LNK < m∠NLM By the Hinge Theorem, KL < MN. C Find the range of values for z. Step 1 Compare the side lengths in △TUV and △TWV. TV = TV VU = VW TU < TW By the Converse of the Hinge Theorem, m∠UVT < m∠WVT. 6z - 3 < 45 Substitute the given values. z < 8 Add 3 to both sides and divide both sides by 6. Step 2 Since ∠UVT is in a triangle, m∠UVT > 0°. 6z - 3 > 0 Substitute the given value. z > 0.5 Add 3 to both sides and divide both sides by 6. Step 3 Combine the inequalities. The range of values for z is 0.5

< z < 8. Compare the given measures. 1a. m∠EGH and m∠EGF 1b. BC and AB E X A M P L E 2 Entertainment Application The angle of the swings in a circular swing ride changes with the speed of the ride. The diagram shows the position of one swing at two different speeds. Which rider is farther from the base of the swing tower? Explain. The height of the tower and the length of the cable holding the chair are the same in both triangles. The angle formed by the swing in position A is smaller than the angle formed by the swing in position B. So rider B is farther from the base of the tower than rider A by the Hinge Theorem. 5- 6 Inequalities in Two Triangles 341 341 ��BAge07sec05L06002_A 2. When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain. E X A M P L E 3 Proving Triangle Relationships Write a two-column proof. ̶̶ NL Given: Prove: KM > NM ̶̶ KL ≅ Proof: ̶̶ KL ≅ ̶̶̶ LM ≅ ̶̶ NL ̶̶̶ LM 1. 2. Statements Reasons 1. Given 2. Reflex. Prop. of ≅ 3. m∠KLM = m∠NLM + m∠KLN 3. ∠ Add. Post. 4. m∠KLM > m∠NLM 4. Comparison Prop. of Inequal. 5. KM > NM 5. Hinge Thm. Write a two-column proof. 3a. Given: C is the midpoint of ̶̶ BD. m∠1 = m∠2 m∠3 > m∠4 Prove: AB > ED 3b. Given: ∠SRT ≅ ∠STR TU > RU Prove: m∠TSU > m∠RSU THINK AND DISCUSS 1. Describe a real-world object that shows the Hinge Theorem or its converse. 2. Can you make a conclusion about the triangles shown at right by applying the Hinge Theorem? Explain. 3. GET ORGANIZED Copy and complete

the graphic organizer. In each box, use the given triangles to write a statement for the theorem. 342 342 Chapter 5 Properties and Attributes of Triangles �� 5-6 Exercises Exercises KEYWORD: MG7 5-6 KEYWORD: MG7 Parent GUIDED PRACTICE Compare the given measures. p. 341 1. AC and XZ 2. m∠SRT and m∠QRT 3. KL and KN Find the range of values for x. 4. 5. 6. Health A therapist can take measurements to p. 341 gauge the flexibility of a patient’s elbow joint. In which position is the angle measure at the elbow joint greater? Explain. 342 Independent Practice For See Exercises Example 9–14 15 16 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 8. Write a two-column proof. Given: ̶̶ FH is a median of △DFG. m∠DHF > m∠GHF Prove: DF > GF PRACTICE AND PROBLEM SOLVING Compare the given measures. 9. m∠DCA and m∠BCA 10. m∠GHJ and m∠KLM 11. TU and SV Find the range of values for z. 12. 13. 14. 5- 6 Inequalities in Two Triangles 343 343 �� 15. Industry The operator of a backhoe changes the distance between the cab and the bucket by changing the angle formed by the arms. In which position is the distance from the cab to the bucket greater? Explain. �� 16. Write a two-column proof. ̶̶̶ MQ, JQ > NP ̶̶ KP ≅ ̶̶̶ NM, ̶̶ JK ≅ Given: Prove: m∠K > m∠M 17. Critical Thinking ABC is an isosceles triangle with base triangle with base ̶̶ YZ. Given that ̶̶ AB ≅ ̶̶ BC. XYZ is an is

osceles ̶̶ XY and m∠A = m∠X, compare BC and YZ. Compare. Write <, >, or =. 18. m∠QRP m∠SRP 19. m∠QPR m∠QRP 20. m∠PRS m∠RSP 21. m∠RSP m∠RPS 22. m∠QPR m∠RPS 23. m∠PSR m∠PQR Make a conclusion based on the Hinge Theorem or its converse. (Hint : Draw a sketch.) ̶̶ DE, ̶̶ RT. The endpoints of 25. △RST is isosceles with base 24. In △ABC and △DEF, ̶̶ BC ≅ ̶̶ AB ≅ ̶̶ EF, m∠B = 59°, and m∠E = 47°. ̶̶ SV are vertex S and a point V on ̶̶ RT. RV = 4, and TV = 5. 26. In △GHJ and △KLM, ̶̶̶ GH ≅ ̶̶ KL, and ̶̶ GJ ≅ ̶̶̶ KM. ∠G is a right angle, and ∠K is an acute angle. 27. In △XYZ, ̶̶̶ XM is the median to ̶̶ YZ, and YX > ZX. 28. Write About It The picture shows a door hinge in two different positions. Use the picture to explain why Theorem 5-6-1 is called the Hinge Theorem. 29. Write About It Compare the Hinge Theorem to the SAS Congruence Postulate. How are they alike? How are they different? 30. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The solid lines in the figure show an airline’s routes between four cities. a. A traveler wants to fly from Jackson (J) to Shelby (S), but there is no direct flight between these cities. Given that m∠NSJ < m∠HSJ, should the traveler first fly to Newton Springs (N) or to Hollis (H) if he wants to minimize the number of miles flown? Why? b. The distance from Shelby (S) to Jackson (J) is 182 mi. What is

the minimum number of miles the traveler will have to fly? 344 344 Chapter 5 Properties and Attributes of Triangles �� 31. ̶̶ ML is a median of △JKL. Which inequality best describes the range of values for x? x > 2 x > 10 < 10 32. ̶̶ DC is a median of △ABC. Which of the following statements is true? BC < AC BC > AC AD = DB DC = AB 33. Short Response Two groups start hiking from the same camp. Group A hikes 6.5 miles due west and then hikes 4 miles in the direction N 35° W. Group B hikes 6.5 miles due east and then hikes 4 miles in the direction N 45° E. At this point, which group is closer to the camp? Explain. CHALLENGE AND EXTEND 34. Multi-Step In △XYZ, XZ = 5x + 15, XY = 8x - 6, and m∠XVZ > m∠XVY. Find the range of values for x. 35. Use these steps to write a paragraph proof of the Hinge Theorem. ̶̶ BC ≅ ̶̶ AB ≅ ̶̶ DE, Given: ̶̶ EF, m∠ABC > m∠DEF Prove: AC > DF a. Locate P outside △ABC so that ∠ABP ≅ ∠DEF and thus ̶̶ BP ≅ ̶̶ AP ≅ ̶̶ EF. Show that △ABP ≅ △DEF and ̶̶ DF. ̶̶ AC so that Show that △BQP ≅ △BQC and thus ̶̶ BQ bisects ∠PBC. Draw ̶̶ QP ≅ ̶̶ QC. b. Locate Q on ̶̶ QP. c. Justify the statements AQ + QP > AP, AQ + QC = AC, AQ + QC > AP, AC > AP, and AC > DF. SPIRAL REVIEW Find the range and mode, if any, of each set of data. (Previous course) 36. 2, 5, 1, 0.5, 0.75, 2 37. 95, 97, 89, 87, 85, 99 38. 5, 5, 7, 9, 4, 4, 8, 7 For the given information, show that m

ǁ n. State any postulates or theorems used. (Lesson 3-3) 39. m∠2 = (3x + 21) °, m∠6 = (7x + 1) °, x = 5 40. m∠4 = (2x + 34) °, m∠7 = (15x + 27) °, x = 7 Find each measure. (Lesson 5-4) 41. DF 42. BC 43. m∠BFD 5- 6 Inequalities in Two Triangles 345 345 6.5 mi6.5 mi4 mi4 miABThird sketchge07se_c05106009aatopo mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� Simplest Radical Form Algebra When a problem involves square roots, you may be asked to give the answer in simplest radical form. Recall that the radicand is the expression under the radical sign. See Skills Bank page S55 Simplest Form of a Square-Root Expression An expression containing square roots is in simplest form when • the radicand has no perfect square factors other than 1. • the radicand has no fractions. • there are no square roots in any denominator. To simplify a radical expression, remember that the square root of a product is equal to the product of the square roots. Also, the square root of a quotient is equal to the quotient of the square roots. √  ab = √  a · √  b, when a ≥ 0 and, when a ≥ 0 and b > 0 Examples Write each expression in simplest radical form. A √  216 √  216 216 has a perfect-square factor of 36, so the expression is not in simplest radical form. √  (36) (6) Factor the radicand. √  36 · √  6 Product Property of Square Roots 6 √  6 Simplify. Try This TAKS Grades 9–11 Obj. 2 Write each expression in simplest radical form. 3. 10_ √  2 2. √3_ 1

. √720 16 346 346 Chapter 5 Properties and Attributes of Triangles ) There is a square root in the denominator, so the expression is not in simplest radical form. Multiply by a form of 1 to eliminate the square root in the denominator Simplify. Divide. 4. √1_ 3 5. √45 5-7 Hands-on Proof of the Pythagorean Theorem In Lesson 1-6, you used the Pythagorean Theorem to find the distance between two points in the coordinate plane. In this activity, you will build figures and compare their areas to justify the Pythagorean Theorem. Use with Lesson 5-7 TEKS G.8.C Congruence and the geometry of size: derive, extend, and use the Pythagorean Theorem. Also G.9.B Activity 1 Draw a large scalene right triangle on graph paper. Draw three copies of the triangle. On each triangle, label the shorter leg a, the longer leg b, and the hypotenuse c. 2 Draw a square with a side length of b - a. Label each side of the square. 3 Cut out the five figures. Arrange them to make the composite figure shown at right. 4 You can think of this composite figure as being made of the two squares outlined in red. What are the side length and area of the small red square? of the large red square? 5 Use your results from Step 4 to write an algebraic expression for the area of the composite figure. 6 Now rearrange the five figures to make a single square with side length c. Write an algebraic expression for the area of this square. Try This 1. Since the composite figure and the square with side length c are made of the same five shapes, their areas are equal. Write and simplify an equation to represent this relationship. What conclusion can you make? 2. Draw a scalene right triangle with different side lengths. Repeat the activity. Do you reach the same conclusion? 5- 7 Geometry Lab 347 347 5-7 The Pythagorean Theorem TEKS G.8.C Congruence and the geometry of size: derive, extend, and use the Pythagorean Theorem. Also G.1.B, G.5.B, G.5.D, G.11.C Objectives Use the Pythagorean Theorem and its converse to solve problems.

Use Pythagorean inequalities to classify triangles. Vocabulary Pythagorean triple Why learn this? You can use the Pythagorean Theorem to determine whether a ladder is in a safe position. (See Example 2.) The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. For more on the Pythagorean Theorem, see the Theorem Builder on page xxvi. � � � The Pythagorean Theorem is named for the Greek mathematician Pythagoras, who lived in the sixth century B.C.E. However, this relationship was known to earlier people, such as the Babylonians, Egyptians, and Chinese. There are many different proofs of the Pythagorean Theorem. The one below uses area and algebra. PROOF PROOF Pythagorean Theorem Given: A right triangle with leg lengths a and b and hypotenuse of length c Prove: a 2 + b 2 = c 2 The area A of a square with side length s is given by the formula A = s 2. The area A of a triangle with base b and height h is given by the formula A = 1 __ bh. 2 Proof: Arrange four copies of the triangle as shown. The sides of the triangles form two squares. The area of the outer square is (a + b) 2. The area of the inner square is c 2. The area of each blue triangle is 1 __ 2 ab. area of outer square = area of 4 blue triangles + area of inner square (a + b) 2 = 4 ( 1 _ 2 ab) + c 2 a 2 + 2ab + b 2 = 2ab + Substitute the areas. Simplify. Subtract 2ab from both sides. The Pythagorean Theorem gives you a way to find unknown side lengths when you know a triangle is a right triangle. 348 348 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form 52 = x 2 √  52 = x Pythagorean Theorem Substitute 6 for a, 4 for b, and x for c. Simplify. Find the positive square root. x = √ �

�� (4) (13) = 2 √  13 Simplify the radicalx - 1) 2 = x 2 Pythagorean Theorem Substitute 5 for a, x - 1 for b, 25 + x 2 - 2x + 1 = x 2 Multiply. and x for c. -2x + 26 = 0 Combine like terms. 26 = 2x x = 13 Add 2x to both sides. Divide both sides by 2. Find the value of x. Give your answer in simplest radical form. 1a. 1b. E X A M P L E 2 Safety Application To prevent a ladder from shifting, safety experts recommend that the ratio of a : b be 4 : 1. How far from the base of the wall should you place the foot of a 10-foot ladder? Round to the nearest inch. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4x is the distance in feet from the top of the ladder to the base of the wall4x) 2 + x 2 = 10 2 17 x 2 = 100 x 2 = 100 _ 17  100 _ 17 x = √ ≈ 2 ft 5 in. Pythagorean Theorem Substitute. Multiply and combine like terms. Divide both sides by 17. Find the positive square root and round it. 2. What if...? According to the recommended ratio, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch. A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple. Common Pythagorean Triples 3, 4, 5 5, 12, 13, 8, 15, 17 7, 24, 25 349 5- 7 The Pythagorean Theorem 349 ��abge07sec05l07002aAB E X A M P L E 3 Identifying Pythagorean Triples Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain 12 2 + b 2 = 15 2 b 2 = 81 b = 9 Pythagorean Theorem Substitute 12 for a and 15 for c. Multiply and subtract 144 from both sides. Find the positive square root. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2,

so they form a Pythagorean triple + 15 2 = c 2 306 = c 2 Pythagorean Theorem Substitute 9 for a and 15 for b. Multiply and add. c = √  306 = 3 √  34 Find the positive square root and simplify. The side lengths do not form a Pythagorean triple because 3 √  34 is not a whole number. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 3a. 3c. 3b. 3d. The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. Theorems 5-7-1 Converse of the Pythagorean Theorem THEOREM HYPOTHESIS CONCLUSION If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. △ABC is a right triangle. a 2 + b 2 = c 2 You will prove Theorem 5-7-1 in Exercise 45. 350 350 Chapter 5 Properties and Attributes of Triangles �� You can also use side lengths to classify a triangle as acute or obtuse. Theorems 5-7-2 Pythagorean Inequalities Theorem In △ABC, c is the length of the longest side. If c 2 > a 2 + b 2, then △ABC is an obtuse triangle. If c 2 < a 2 + b 2, then △ABC is an acute triangle. To understand why the Pythagorean inequalities are true, consider △ABC. If c 2 = a 2 + b 2, then △ABC is a right triangle by the Converse of the Pythagorean Theorem. So m∠C = 90°. If c 2 > a 2 + b 2, then c has increased. By the Converse of the Hinge Theorem, m∠C has also increased. So m∠C > 90°. If c 2 < a 2 + b 2, then c has decreased. By the Converse of the Hinge Theorem, m∠C has also decreased. So m∠C < 90°. E X A M P L E 4 Classifying Triangles By the Triangle Inequality Theorem

, the sum of any two side lengths of a triangle is greater than the third side length. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. A 8, 11, 13 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 8, 11, and 13 can be the side lengths of a triangle. Step 2 Classify the triangle. c 2 ≟ a 2 + b 2 13 2 ≟ 8 2 + 11 2 169 ≟ 64 + 121 169 < 185 Compare c 2 to a 2 + b 2. Substitute the longest side length for c. Multiply. Add and compare. Since c 2 < a 2 + b 2, the triangle is acute. B 5.8, 9.3, 15.6 Step 1 Determine if the measures form a triangle. Since 5.8 + 9.3 = 15.1 and 15.1 ≯ 15.6, these cannot be the side lengths of a triangle. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 4a. 7, 12, 16 4b. 11, 18, 34 4c. 3.8, 4.1, 5.2 351 5- 7 The Pythagorean Theorem 351 �� THINK AND DISCUSS 1. How do you know which numbers to substitute for c, a, and b when using the Pythagorean Inequalities? 2. Explain how the figure at right demonstrates the Pythagorean Theorem. 3. List the conditions that a set of three numbers must satisfy in order to form a Pythagorean triple. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, summarize the Pythagorean relationship. 5-7 Exercises Exercises KEYWORD: MG7 5-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Do the numbers 2.7, 3.6, and 4.5 form a Pythagorean triple? Explain why or why not Find the value of x. Give your answer in simplest radical form. p. 349 2. 3. 4. 349 5. Computers The size of a computer monitor is usually given by the length of its diagonal. A monitor’s aspect ratio is the ratio of its

width to its height. This monitor has a diagonal length of 19 inches and an aspect ratio of 5 : 4. What are the width and height of the monitor? Round to the nearest tenth of an inch. 350 Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 6. 7. 8. 351 Multi-Step Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 9. 7, 10, 12, 1 3 _, 3 1 _ 12. 1 1 _ 4 4 2 10. 9, 11, 15 13. 5.9, 6, 8.4 11. 9, 40, 41 14. 11, 13, 7 √  6 352 352 Chapter 5 Properties and Attributes of Triangles �� PRACTICE AND PROBLEM SOLVING Find the value of x. Give your answer in simplest radical form. 15. 16. 17. 18. Safety The safety rules for a playground state that the height of the slide and the distance from the base of the ladder to the front of the slide must be in a ratio of 3 : 5. If a slide is about 8 feet long, what are the height of the slide and the distance from the base of the ladder to the front of the slide? Round to the nearest inch. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 19. 20. 21. Multi-Step Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 22. 10, 12, 15, 2, 2 1 _ 25. 1 1 _ 2 2 23. 8, 13, 23 26. 0.7, 1.1, 1.7 24. 9, 14, 17 27. 7, 12, 6 √  5 28. Surveying It is believed that surveyors in ancient Egypt laid out right angles using a rope divided into twelve sections by eleven equally spaced knots. How could the surveyors use this rope to make a right angle? 29. /////ERROR ANALYSIS///// Below are two solutions for finding x. Which is incorrect? Explain the error. Independent

Practice For See Exercises Example 15–17 18 19–21 22–27 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 Surveying Ancient Egyptian surveyors were referred to as rope-stretchers. The standard surveying rope was 100 royal cubits. A cubit is 52.4 cm long. Find the value of x. Give your answer in simplest radical form. 30. 31. 33. 34. 32. 35. 353 5- 7 The Pythagorean Theorem 353 �� 36. Space Exploration The International Space Station orbits at an altitude of about 250 miles above Earth’s surface. The radius of Earth is approximately 3963 miles. How far can an astronaut in the space station see to the horizon? Round to the nearest mile. 37. Critical Thinking In the proof of the Pythagorean Theorem on page 348, how do you know the outer figure is a square? How do you know the inner figure is a square? Multi-Step Find the perimeter and the area of each figure. Give your answer in simplest radical form. 38. 41. 39. 42. 40. 43. 44. Write About It When you apply both the Pythagorean Theorem and its converse, you use the equation a 2 + b 2 = c 2. Explain in your own words how the two theorems are different. 45. Use this plan to write a paragraph proof of the Converse of the Pythagorean Theorem. Given: △ABC with a 2 + b 2 = c 2 Prove: △ABC is a right triangle. Plan: Draw △PQR with ∠R as the right angle, leg lengths of a and b, and a hypotenuse of length x. By the Pythagorean Theorem, a 2 + b 2 = x 2. Use substitution to compare x and c. Show that △ABC ≅ △PQR and thus ∠C is a right angle. 46. Complete these steps to prove the Distance Formula. Given: J ( x

1, y 1 ) and K ( x 2, y 2 ) with x 1 ≠ x 2 and y 1 ≠ y 2 Prove: JK = √  ( ̶̶ a. Locate L so that JK is the hypotenuse of right △JKL. What are the coordinates of L? b. Find JL and LK. c. By the Pythagorean Theorem, JK 2 = JL 2 + LK 2. Find JK. 47. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes between four cities. a. A traveler wants to go from Sanak (S) to Manitou (M). To minimize the total number of miles traveled, should she first fly to King City (K) or to Rice Lake (R)? b. The airline decides to offer a direct flight from Sanak (S) to Manitou (M). Given that the length of this flight is more than 1360 mi, what can you say about m∠SRM? 354 354 Chapter 5 Properties and Attributes of Triangles �� 48. Gridded Response ̶̶ KX, ̶̶ LX, and ̶̶̶ MX are the perpendicular bisectors of △GHJ. Find GJ to the nearest tenth of a unit. 49. Which number forms a Pythagorean triple with 24 and 25? 1 7 26 49 50. The lengths of two sides of an obtuse triangle are 7 meters and 9 meters. Which could NOT be the length of the third side? 4 meters 5 meters 11 meters 12 meters 51. Extended Response The figure shows the first six triangles in a pattern of triangles. a. Find PA, PB, PC, PD, PE, and PF in simplest radical form. b. If the pattern continues, what would be the length of the hypotenuse of the ninth triangle? Explain your answer. c. Write a rule for finding the length of the hypotenuse of the nth triangle in the pattern. Explain your answer. CHALLENGE AND EXTEND 52. Algebra

Find all values of k so that (-1, 2), (-10, 5), and (-4, k) are the vertices of a right triangle. 53. Critical Thinking Use a diagram of a right triangle to explain why a + b > √  a 2 + b 2 for any positive numbers a and b. 54. In a right triangle, the leg lengths are a and b, and the length of the altitude to the hypotenuse is h. Write an expression for h in terms of a and b. (Hint: Think of the area of the triangle.) 55. Critical Thinking Suppose the numbers a, b, and c form a Pythagorean triple. Is each of the following also a Pythagorean triple? Explain. a. a + 1, b + 1, c + 1 c. 2a, 2b, 2c d SPIRAL REVIEW Solve each equation. (Previous course) 56. (4 + x) 12 - (4x + 1) 6 = 0 57. 2x - 5 _ 3 = x 58. 4x + 3 (x + 2) = -3 (x + 3) Write a coordinate proof. (Lesson 4-7) 59. Given: ABCD is a rectangle with A (0, 0), B (0, 2b), C (2a, 2b), and D (2a, 0). M is the midpoint of ̶̶ AC. Prove: AM = MB Find the range of values for x. (Lesson 5-6) 60. 61. 355 5- 7 The Pythagorean Theorem 355 �� 5-8 Applying Special Right Triangles TEKS G.5.D Geometric patterns: identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90)... Objectives Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°-60°-90° triangles. Also G.3.B, G.5.A, G.7.A Who uses this? You can use properties of special right triangles to calculate the correct size of a bandana for your dog. (See Example 2.) A diagonal of a square divides it into two congruent isosceles

right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle. A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° triangle √  Pythagorean Theorem Substitute the given values. Simplify. Find the square root of both sides. Simplify. Theorem 5-8-1 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √  2. AC = BC = ℓ AB = Finding Side Lengths in a 45°-45°-90° Triangle Find the value of x. Give your answer in simplest radical form. A By the Triangle Sum Theorem, the measure of the third angle of the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 7. x = 7 √  2 Hypotenuse = leg √  2 356 356 Chapter 5 Properties and Attributes of Triangles �� Find the value of x. Give your answer in simplest radical form. B The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 3. 3 = x √  2 Hypotenuse = leg √  Divide both sides by √  2. Rationalize the denominator. Find the value of x. Give your answer in simplest radical form. 1a. 1b. E X A M P L E 2 Craft Application � � � � Tessa wants to make a bandana for her dog by folding a square of cloth into a 45°-45°-90° triangle. Her dog’s neck has a circumference of about 32 cm. The folded bandana needs to be an extra 16 cm long so Tessa can tie it around her dog’s neck. What should the side length of the square be? Round to the nearest

centimeter. � � �� Tessa needs a 45°-45°-90° triangle with a hypotenuse of 48 cm. 48 = ℓ √  2 ℓ = 48 _ √  2 Divide by √  2 and round. Hypotenuse = leg √  2 ≈ 34 cm 2. What if...? Tessa’s other dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of the other dog’s neck to be? Round to the nearest centimeter. A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths. Draw an altitude in △PQR. Since △PQS ≅ △RQS, ̶̶ PS ≅ the Pythagorean Theorem to find y. ̶̶ RS. Label the side lengths in terms of x, and use 2x ) = √  3 x 2 y = x √  3 Pythagorean Theorem Substitute x for a, y for b, and 2x for c. Multiply and combine like terms. Find the square root of both sides. Simplify. 5- 8 Applying Special Right Triangles 357 357 �� Theorem 5-8-2 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √  3. AC = s AB = 2s BC = Finding Side Lengths in a 30°-60°-90° Triangle Find the values of x and y. Give your answers in simplest radical form. A B 16 = 2x Hypotenuse = 2 (shorter leg) Divide both sides by 2. Longer leg = (shorter leg) √  3 Substitute 8 for x. 11 = x √  3 Longer leg = (shorter leg) √  3 If two angles of a triangle are not congruent, the shorter side lies opposite the smaller angle. = x

= x 11 _ √  3 11 √  3 _ 3 y = 2x y = 2 ( ) _ 11 √  3 3 y = 22 √  3 _ 3 Divide both sides by √  3. Rationalize the denominator. Hypotenuse = 2 (shorter leg) Substitute 11 √  3 _ 3 for x. Simplify. Find the values of x and y. Give your answers in simplest radical form. 3a. 3c. 3b. 3d. 358 358 Chapter 5 Properties and Attributes of Triangles �� 30°-60°-90° Triangles To remember the side relationships in a 30°-60°-90° triangle, I draw a simple “1-2- √  3 ” triangle like this. 2 = 2 (1), so hypotenuse = 2 (shorter leg1), so longer leg = √  3 (shorter leg). Marcus Maiello Johnson High School E X A M P L E 4 Using the 30°-60°-90° Triangle Theorem The frame of the clock shown is an equilateral triangle. The length of one side of the frame is 20 cm. Will the clock fit on a shelf that is 18 cm below the shelf above it? Step 1 Divide the equilateral triangle into two 30°-60°-90° triangles. The height of the frame is the length of the longer leg. Step 2 Find the length x of the shorter leg. Hypotenuse = 2(shorter leg) Divide both sides by 2. 20 = 2x 10 = x Step 3 Find the length h of the longer leg. h = 10 √  3 ≈ 17.3 cm Longer leg = (shorter leg) √  3 The frame is approximately 17.3 centimeters tall. So the clock will fit on the shelf. 4. What if…? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth. THINK AND DISCUSS 1. Explain why an isosceles right triangle is a 45°-45°-90° triangle. 2. Describe how finding x in triangle I is different from finding

x in triangle II. I. II. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, sketch the special right triangle and label its side lengths in terms of s. 5- 8 Applying Special Right Triangles 359 359 �� 5-8 Exercises Exercises GUIDED PRACTICE Find the value of x. Give your answer in simplest radical form. p. 356 1. 2. 3. KEYWORD: MG7 5-8 KEYWORD: MG7 Parent. 357 4. Transportation The two arms of the railroad sign are perpendicular bisectors of each other. In Pennsylvania, the lengths marked in red must be 19.5 inches. What is the distance labeled d? Round to the nearest tenth of an inch Find the values of x and y. Give your answers in simplest radical form. p. 358 5. 6. 7. 359 8. Entertainment Regulation billiard balls are 2 1 __ 4 inches in diameter. The rack used to group 15 billiard balls is in the shape of an equilateral triangle. What is the approximate height of the triangle formed by the rack? Round to the nearest quarter of an inch. Independent Practice For See Exercises Example 9–11 12 13–15 16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING Find the value of x. Give your answer in simplest radical form. 9. 10. 11. 12. Design This tabletop is an isosceles 12. right triangle. The length of the front edge of the table is 48 inches. What is the length w of each side edge? Round to the nearest tenth of an inch. � � �� Find the value of x and y. Give your answers in simplest radical form. 13. 14. 15. 360 360 Chapter 5 Properties and Attributes of Triangles �� Pets 16. Pets A dog walk is used in dog agility 16. competitions. In this dog walk, each ramp makes an angle of 30° with the ground. a. How long is one ramp? b. b. How long is the

entire dog walk, including both ramps? Multi-Step Find the perimeter and area of each figure. Give your answers in simplest radical form. 17. 17. a 45°-45°-90° triangle with hypotenuse length 12 inches 18. 18. a 30°-60°-90° triangle with hypotenuse length 28 centimeters Agility courses test the skill of both the dog and the dog’s handler. Dog agility competitions in the United States are regulated by the U.S. Dog Agility Association, headquartered in Garland, Texas. 19. 19. a square with diagonal length 18 meters 20. an equilateral triangle with side length 4 feet 21. an equilateral triangle with height 30 yards 22. Estimation The triangle loom is made from wood strips shaped into a 45°-45°-90° triangle. Pegs are placed every 1 __ 2 inch along the hypotenuse and every 1 __ 4 inch along each leg. Suppose you make a loom with an 18-inch hypotenuse. Approximately how many pegs will you need? 23. Critical Thinking The angle measures of a triangle are in the ratio 1 : 2 : 3. Are the side lengths also in the ratio 1 : 2 : 3? Explain your answer. Find the coordinates of point P under the given conditions. Give your answers in simplest radical form. 24. △PQR is a 45°-45°-90° triangle with vertices Q (4, 6) and R (-6, -4), and m∠P = 90°. P is in Quadrant II. 25. △PST is a 45°-45°-90° triangle with vertices S (4, -3) and T (-2, 3), and m∠S = 90°. P is in Quadrant I. 26. △PWX is a 30°-60°-90° triangle with vertices W (-1, -4) and X (4, -4), and m∠W = 90°. P is in Quadrant II. 27. △PYZ is a 30°-60°-90° triangle with vertices Y (-7, 10) and Z (5, 10), and m∠Z = 90°. P is in Quadrant IV. 28. Write About It Why do you think 30°-60°-90° triangles and 45°-45°-90° triangles are

called special right triangles? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes among four cities. The airline offers one frequent-flier mile for each mile flown (rounded to the nearest mile). How many frequent-flier miles do you earn for each flight? a. Nelson (N) to Belton (B) b. Idria (I) to Nelson (N) c. Belton (B) to Idria (I) 5- 8 Applying Special Right Triangles 361 361 30°30°12 ft4.5 ftge07se_c05l08007aAB�� 30. Which is a true statement? AB = BC √  2 AB = BC √  3 AC = BC √  3 AC = AB √  2 31. An 18-foot pole is broken during a storm. The top of the pole touches the ground 12 feet from the base of the pole. How tall is the part of the pole left standing? 5 feet 6 feet 13 feet 22 feet �� 32. The length of the hypotenuse of an isosceles right triangle is 24 inches. What is the length of one leg of the triangle, rounded to the nearest tenth of an inch? 13.9 inches 17.0 inches 33.9 inches 41.6 inches 33. Gridded Response Find the area of the rectangle to the nearest tenth of a square inch. CHALLENGE AND EXTEND Multi-Step Find the value of x in each figure. 34. 35. 36. Each edge of the cube has length e. a. Find the diagonal length d when e = 1, e = 2, and e = 3. Give the answers in simplest radical form. b. Write a formula for d for any positive value of e. 37. Write a paragraph proof to show that the altitude to the hypotenuse of a 30°-60°-90° triangle divides the hypotenuse into two segments, one of which is 3 times as long as the other. SPIRAL REVIEW Rewrite each function in the form y = a (x - h) 2 - k and find the axis of symmetry. (Previous course) 38. y = x 2 + 4x 39. y = x 2 - 10x -2 40. y = x 2 + 7x +15 Classify each

triangle by its angle measures. (Lesson 4-1) 41. △ ADB 42. △BDC 43. △ ABC Use the diagram for Exercises 44–46. (Lesson 5-1) 44. Given that PS = SR and m∠PSQ = 65°, find m∠PQR. 45. Given that UT = TV and m∠PQS = 42°, find m∠VTS. 46. Given that ∠PQS ≅ ∠SQR, SR = 3TU, and PS = 7.5, find TV. 362 362 Chapter 5 Properties and Attributes of Triangles �� 5-8 Graph Irrational Numbers Numbers such as √  2 and √  3 are irrational. That is, they cannot be written as the ratio of two integers. In decimal form, they are infinite nonrepeating decimals. You can round the decimal form to estimate the location of these numbers on a number line, or you can use right triangles to construct their locations exactly. Use with Lesson 5-8 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.5.B Activity 1 Draw a line. Mark two points near the left side of the line and label them 0 and 1. The distance from 0 to 1 is 1 unit. 2 Set your compass to 1 unit and mark increments at 2, 3, 4, and 5 units to construct a number line. 3 Construct a perpendicular to the line through 1. 4 Using your compass, mark 1 unit up from the number line and then draw a right triangle. The legs both have length 1, so by the Pythagorean Theorem, the hypotenuse has a length of √2. 5 Set your compass to the length of the hypotenuse. Draw an arc centered at 0 that intersects the number line at √  2. 6 Repeat Steps 3 through 5, starting at √  2, to construct a segment of length √  3. Try This 1. Sketch the two right triangles from Step 6. Label the side lengths and use the Pythagorean Theorem to show why the construction is correct. 2. Construct √4 and verify that

it is equal to 2. 3. Construct √5 through √9 and verify that √9 is equal to 3. 4. Set your compass to the length of the segment from 0 to √2. Mark off another segment of length √2 to show that √8 is equal to 2 √2. 5- 8 Geometry Lab 363 363 �� SECTION 5B Relationships in Triangles Fly Away! A commuter airline serves the four cities of Ashton, Brady, Colfax, and Dumas, located at points A, B, C, and D, respectively. The solid lines in the figure show the airline’s existing routes. The airline is building an airport at H, which will serve as a hub. This will add four new routes to their schedule: ̶̶ CH, and ̶̶ BH, ̶̶ AH, ̶̶̶ DH. 1. The airline wants to locate the airport so that the combined distance to the cities (AH + BH + CH + DH) is as small as possible. Give an indirect argument to explain why the airline should locate the airport at the ̶̶ AC and intersection of the diagonals point X inside quadrilateral ABCD results in a smaller combined distance. Then consider how AX + CX compares to AH + CH.) ̶̶ BD. (Hint: Assume that a different 2. Currently, travelers who want to go from Ashton to Colfax must first fly to Brady. Once the airport is built, they will fly from Ashton to the new airport and then to Colfax. How many miles will this save compared to the distance of the current trip? 3. Currently, travelers who want to go from Brady to Dumas must first fly to Colfax. Once the airport is built, they will fly from Brady to the new airport and then to Dumas. How many miles will this save? 4. Once the airport is built, the airline plans to serve a meal only on its longest flight. On which route should they serve the meal? How do you know that this route is the longest? 364 364 Chapter 5 Properties and Attributes of Triangles �� SECTION 5B Quiz for Lessons 5-5 Through 5-8 5-5 Indirect Proof and Inequalities in One Triangle 1. Write an

indirect proof that the supplement of an acute angle cannot be an acute angle. 2. Write the angles of △KLM in order from smallest to largest. 3. Write the sides of △DEF in order from shortest to longest. Tell whether a triangle can have sides with the given lengths. Explain. 4. 8.3, 10.5, 18.8 5. 4s, s + 10, s 2, when s = 4 6. The distance from Kara’s school to the theater is 9 km. The distance from her school to the zoo is 16 km. If the three locations form a triangle, what is the range of distances from the theater to the zoo? 5-6 Inequalities in Two Triangles 7. Compare PR and SV. 8. Compare m∠KJL and 9. Find the range of m∠MJL. values for x. 5-7 The Pythagorean Theorem 10. Find the value of x. Give the answer in simplest radical form. 11. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 12. Tell if the measures 10, 12, and 16 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 13. A landscaper wants to place a stone walkway from one corner of the rectangular lawn to the opposite corner. What will be the length of the walkway? Round to the nearest inch. 5-8 Applying Special Right Triangles 14. A yield sign is an equilateral triangle with a side length of 36 inches. �� What is the height h of the sign? Round to the nearest inch. Find the values of the variables. Give your answers in simplest radical form. 15. 16. 17. �� Ready to Go On? 365 365 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary altitude of a triangle........ 316 equidistant................. 300 median of a triangle........ 314 centroid of a triangle........ 314 incenter of a triangle

........ 309 midsegment of a triangle.... 322 circumcenter of a triangle... 307 indirect proof............... 332 orthocenter of a triangle.... 316 circumscribed.............. 308 inscribed................... 309 point of concurrency........ 307 concurrent................. 307 locus....................... 300 Pythagorean triple.......... 349 Complete the sentences below with vocabulary words from the list above. 1. A point that is the same distance from two or more objects is? from the objects. ̶̶̶̶ 2. A? is a segment that joins the midpoints of two sides of the triangle. ̶̶̶̶ 3. The point of concurrency of the angle bisectors of a triangle is the?. ̶̶̶̶ 4. A? is a set of points that satisfies a given condition. ̶̶̶̶ 5-1 Perpendicular and Angle Bisectors (pp. 300–306) E X A M P L E S Find each measure. ■ JL ̶̶̶ MK and ̶̶ JM ≅ ̶̶̶ ML is the Because ̶̶ ̶̶̶ ML ⊥ JK, perpendicular bisector of ̶̶ JK. EXERCISES Find each measure. 5. BD 6. YZ TEKS G.3.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B JL = KL ⊥ Bisector Thm. JL = 7.9 Substitute 7.9 for KL. ■ m∠PQS, given that m∠PQR = 68° ̶̶ ̶̶ SP ⊥ QP, Since SP = SR, ̶̶ ̶̶  QS bisects QR, SR ⊥ and ∠PQR by the Converse of the Angle Bisector Theorem.

m∠PQR m∠PQS = 1 _ 2 m∠PQS = 1 _ (68°) = 34° 2 Def. of ∠ bisector Substitute 68° for m∠PQR. 7. HT 8. m∠MNP Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. A (-4, 5), B (6, -5) 10. X (3, 2), Y (5, 10) Tell whether the given information allows you to conclude that P is on the bisector of ∠ABC. 11. 12. 366 366 Chapter 5 Properties and Attributes of Triangles �� 5-2 Bisectors of Triangles (pp. 307–313) TEKS G.2.A, G.2.B, G.3.B, G.7.A, G.7.B E X A M P L E S ̶̶ ̶̶ FG EG, and ■ ̶̶ DG, are the perpendicular bisectors of △ABC. Find AG. G is the circumcenter of △ABC. By the Circumcenter Theorem, G is equidistant from the vertices of △ABC. AG = CG Circumcenter Thm. AG = 5.1 Substitute 5.1 for CG. ■ ̶̶ ̶̶ QS and RS are angle bisectors of △PQR. Find the ̶̶ PR. distance from S to S is the incenter of △PQR. By the Incenter Theorem, S is equidistant from the sides of △PQR. The distance from S to ̶̶ PR is also 17. distance from S to ̶̶ PQ is 17, so the ̶̶ PY, and EXERCISES ̶̶ PX, perpendicular bisectors of △GHJ. Find each length. ̶̶ PZ are the 13. GY 15. GJ 14. GP 16. PH ̶̶ ̶̶ UA and VA are angle bisectors of △UVW. Find each measure. 17. the distance from ̶̶ UV A to 18. m∠WVA Find the circumcenter of a triangle with the given

vertices. 19. M (0, 6), N (8, 0), O (0, 0) 20. O (0, 0), R (0, -7), S (-12, 0) 5-3 Medians and Altitudes of Triangles (pp. 314–320) TEKS G.2.A, G.2.B, G.3.B, G.7.A, G.7.B, G.7.C E X A M P L E S ■ In △JKL, JP = 42. Find JQ. JP JQ = 2_ 3 JQ = 2_ 3 JQ = 28 (42) Centroid Thm. Substitute 42 for JP. Multiply. ■ Find the orthocenter of △RST with vertices R (-5, 3), S (-2, 5), and T (-2, 0). Since ̶̶ ST is vertical, the equation of the line containing the altitude ̶̶ ST is y = 3. from R to ̶̶ RT = slope of 3 - 0 _ -5 - (-2) = -1 ̶̶ RT is 1. The slope of the altitude to This line must pass through S (-2, 5). y - y 1 = m (x - x 1 ) Point-slope form y - 5 = 1 (x + 2) ⎧ y = 3 ⎨ Solve the system y = x + 7 ⎩ Substitution to find that the EXERCISES In △DEF, DB = 24.6, and EZ = 11.6. Find each length. 21. DZ 22. ZB 23. ZC 24. EC Find the orthocenter of a triangle with the given vertices. 25. J (-6, 7), K (-6, 0), L (-11, 0) 26. A (1, 2), B (6, 2), C (1, -8) 27. R (2, 3), S (7, 8), T (8, 3) 28. X (-3, 2), Y (5, 2), Z (3, -4) 29. The coordinates of a triangular piece of a mobile are (0, 4), (3, 8), and (6, 0). The piece will hang from a chain so that

it is balanced. At what coordinates should the chain be attached? coordinates of the orthocenter are (-4, 3). Study Guide: Review 367 367 �� TEKS G.2.A, G.2.B, G.3.B, G.5.A, G.7.B, G.9.B 5-4 The Triangle Midsegment Theorem (pp. 322–327) E X A M P L E S Find each measure. ■ NQ By the △ Midsegment Thm., NQ = 1 _ 2 KL = 45.7. ■ m∠NQM ̶̶̶ ̶̶ ML NP ǁ m∠NQM = m∠PNQ m∠NQM = 37° △ Midsegment Thm. Alt. Int.  Thm. Substitution EXERCISES Find each measure. 30. BC 31. XZ 32. XC 33. m∠BCZ 34. m∠BAX 35. m∠YXZ 36. The vertices of △GHJ are G (-4, -7), H (2, 5), and J (10, -3). V is the midpoint of ̶̶ HJ. Show that W is the midpoint of and VW = 1 __ 2 GJ. ̶̶̶ GH, and ̶̶ ̶̶̶ GJ VW ǁ 5-5 Indirect Proof and Inequalities in One Triangle (pp. 332–339) TEKS G.3.B, E X A M P L E S ■ Write the angles of △RST in order from smallest to largest. EXERCISES 37. Write the sides of △ABC in order from shortest to longest. The smallest angle is opposite the shortest side. In order, the angles are ∠S, ∠R, and ∠T. 38. Write the angles of △FGH in order from smallest to largest. G.3.C, G.3.E, G.5.B ■ The lengths of two sides of a triangle are 15 inches and 12 inches. Find the range of possible lengths for the third side. Let s be the length of the third side. s + 15 > 12 s > -3 s + 12

> 15 s > 3 15 + 12 > s 27 > s By the Triangle Inequality Theorem, 3 in. < s < 27 in. 39. The lengths of two sides of a triangle are 13.5 centimeters and 4.5 centimeters. Find the range of possible lengths for the third side. Tell whether a triangle can have sides with the given lengths. Explain. 40. 6.2, 8.1, 14.2 41. z, z, 3z, when z = 5 42. Write an indirect proof that a triangle cannot have two obtuse angles. 5-6 Inequalities in Two Triangles (pp. 340–345) TEKS G.3.B, G.3.E E X A M P L E S Compare the given measures. ■ KL and ST KJ = RS, JL = RT, and m∠J > m∠R. By the Hinge Theorem, KL > ST. ■ m∠ZXY and m∠XZW XY = WZ, XZ = XZ, and YZ < XW. By the Converse of the Hinge Theorem, m∠ZXY < m∠XZW. 368 368 Chapter 5 Properties and Attributes of Triangles EXERCISES Compare the given measures. 43. PS and RS 44. m∠BCA and m∠DCA Find the range of values for n. 46. 45. �� 5-7 The Pythagorean Theorem (pp. 348–355) TEKS G.1.B, G.5.B, G.5.D, G.8.C, G.11.C E X A M P L E S EXERCISES ■ Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 62 + 32 = x2 45 = x 2 x = 3 √5 Pyth. Thm. Substitution Simplify. Find the positive square root and simplify. ■ Find the missing side length. Tell if the sides form a Pythagorean triple. Explain1.6)2 = 22 a 2 = 1.44 a = 1.2 Pyth. Thm. Substitution Solve for a 2

. Find the positive square root. The side lengths do not form a Pythagorean triple because 1.2 and 1.6 are not whole numbers. Find the value of x. Give your answer in simplest radical form. 47. 48. Find the missing side length. Tell if the sides form a Pythagorean triple. Explain. 49. 50. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 51. 9, 12, 16 52. 11, 14, 27 53. 1.5, 3.6, 3.9 54. 2, 3.7, 4.1 5-8 Applying Special Right Triangles (pp. 356–362) TEKS G.3.B, G.5.A, G.5.D, G.7.A E X A M P L E S EXERCISES Find the values of the variables. Give your answers in simplest radical form. ■ ■ ■ This is a 45°-45°-90° triangle. x = 19 √  2 Hyp. = leg √  2 This is a 45°-45°-90° triangle. 15 = x √  2 Hyp. = leg √  2 Divide both sides by √  2. Rationalize the denominator. This is a 30°-60°-90° triangle. 22 = 2x Hyp. = 2(shorter leg) = x 15 _ √  2 15 √  2 _ 2 = x 11 = x y = 11 √  3 Divide both sides by 2. Longer leg = (shorter leg) √  3 Find the values of the variables. Give your answers in simplest radical form. 55. 56. 57. 59. 58. 60. Find the value of each variable. Round to the nearest inch. 61. 62. Study Guide: Review 369 369 �� Find each measure. 1. KL 2. m∠WXY 3. BC 4. ̶̶̶ NQ, and ̶̶ PQ are the ̶̶̶ MQ, perpendicular bisectors of △RST. Find RS and RQ

. 5. ̶̶ FG are angle ̶̶ EG and bisectors of △DEF. Find m∠GEF and the ̶̶ DF. distance from G to 6. In △XYZ, XC = 261, and ZW = 118. Find XW, BW, and BZ. 7. Find the orthocenter of △JKL with vertices J (-5, 2), K (-5, 10), and L (1, 4). 8. In △GHJ at right, find PR, GJ, and m∠GRP. 9. Write an indirect proof that two obtuse angles cannot form a linear pair. 10. Write the angles of △BEH in order from smallest to largest. 11. Write the sides of △RTY in order from shortest to longest. 12. The distance from Arville to Branton is 114 miles. The distance from Branton to Camford is 247 miles. If the three towns form a triangle, what is the range of distances from Arville to Camford? 13. Compare m∠SPV and m∠ZPV. 14. Find the range of values for x. 15. Find the missing side length in the triangle. Tell if the side lengths form a Pythagorean triple. Explain. 16. Tell if the measures 18, 20, and 27 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 17. An IMAX screen is 62 feet tall and 82 feet wide. What is the length of the screen’s diagonal? Round to the nearest inch. Find the values of the variables. Give your answers in simplest radical form. 18. 19. 20. 370 370 Chapter 5 Properties and Attributes of Triangles �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Some questions on the SAT Mathematics Subject Tests require the use of a calculator. You can take the test without one, but it is not recommended. The calculator you use must meet certain criteria. For example, calculators that make noise or have typewriter-like keypads are not allowed. You may want to time yourself as you take this practice test.

It should take you about 6 minutes to complete. If you have both a scientific and a graphing calculator, bring the graphing calculator to the test. Make sure you spend time getting used to a new calculator before the day of the test. 1. In △ABC, m∠C = 2m∠A, and CB = 3 units. What is AB to the nearest hundredth unit? 3. The side lengths of a right triangle are 2, 5, and c, where c > 5. What is the value of c? (A) 1.73 units (B) 4.24 units (C) 5.20 units (D) 8.49 units (E) 10.39 units 2. What is the perimeter of △ABC if D is the ̶̶ AB, E is the midpoint of ̶̶ BC, and midpoint of F is the midpoint of ̶̶ AC? Note: Figure not drawn to scale. (A) 8 centimeters (B) 14 centimeters (C) 20 centimeters (D) 28 centimeters (E) 35 centimeters (A) √  21 (B) √  29 (C) 7 (D) 9 (E) √  145 4. In the triangle below, which of the following CANNOT be the length of the unknown side? (A) 2.2 (B) 6 (C) 12.8 (D) 17.2 (E) 18.1 5. Which of the following points is on the perpendicular bisector of the segment with endpoints (3, 4) and (9, 4)? (A) (4, 2) (B) (4, 5) (C) (5, 4) (D) (6, -1) (E) (7, 4) College Entrance Exam Practice 371 371 �� Any Question Type: Check with a Different Method It is important to check all of your answers on a test. An effective way to do this is to use a different method to answer the question a second time. If you get the same answer with two different methods, then your answer is probably correct. Multiple Choice What are the coordinates of the centroid of △ABC with A (-2, 4), B (4, 6), and C (1, -1)? (1, 5) (2.5, 2.5) (1,

3) (3, 1) Method 1: The centroid of a triangle is the point of concurrency of the medians. Write the equations of two medians and find their point of intersection. Let D be the midpoint of ̶̶ AB and let E be the midpoint of ̶̶ BC. -1,5) The median from C to D contains C (1, -1) and D (1, 5). 6 + (-12.5, 2.5) It is vertical, so its equation is x = 1. The median from A to E contains A (-2, 4) and E (2.5, 2.5). slope of ̶̶ AE = 4 - 2.5 _ -2 - 2.5 = 1.5 _ -4.x + 2) 3 Point-slope form Substitute 4 for y 1, - 1 _ 3 and -2 for x 1. for m, ⎧ x = 1 Solve the system ⎨ y - 4 = - 1 __ (x + 2) ⎩ 3 to find the point of intersection1 + 2) Substitute 1 for x. y = 3 Simplify. The coordinates of the centroid are (1, 3). So choice H is the correct answer. Method 2: To check this answer, use a different method. By the Centroid Theorem, the centroid of a triangle is 2 __ 3 of the distance from each vertex to the midpoint of the ̶̶ CD is vertical with a length of 6 units. 2 __ (6) = 4, opposite side. 3 and the coordinates of the point that is 4 units up from C is (1, 3). This method confirms that choice H is the correct answer. Problem Solving Strategies • Draw a Diagram • Make a Model • Guess and Test • Work Backward • Find a Pattern • Make a Table • Solve a Simpler Problem • Use Logical Reasoning • Use a Venn Diagram • Make an Organized List 372 372 Chapter 5 Properties and Attributes of Triangles �� If you can’t think of a different method to use to check your answer, circle the question and come back to it later. Item C Gridded Response Find the area of the square in square centimeters. Read each test item and answer the questions that follow. Item A Multiple Choice Given that ℓ is the perpendicular bisector of and BC =

6n - 11, what is the value of n? ̶̶ AB, AC = 3n + 1, 5. How can you use special right triangles to answer this question? 6. Explain how you can check your answer by using the Pythagorean Theorem. -4 3 _ 4 4 _ 3 4 1. How can you use the given answer choices to solve this problem? 2. Describe how to solve this problem directly. Item B Multiple Choice Which number forms a Pythagorean triple with 15 and 17? 5 7 8 10 3. How can you use the given answer choices to find the answer? 4. Describe a different method you can use to check your answer. Item D Multiple Choice Which coordinates for point Z form a right triangle with the points X (-8, 4) and Y (0, -2)? Z (4, 4) Z (4, 6) Z (3, 2) Z (8, 4) 7. Explain how to use slope to determine if △XYZ is a right triangle. 8. How can you use the Converse of the Pythagorean Theorem to check your answer? Item E Multiple Choice What are the coordinates of the orthocenter of △RST? (0, 2) (0, 1) (-1, 3) (1, 2) 9. Describe how you would solve this problem directly. 10. How can you use the third altitude of the triangle to confirm that your answer is correct? TAKS Tackler 373 373 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–5 Multiple Choice 1. ̶̶ GJ is a midsegment of △DEF, and midsegment of △GFJ. What is the length of ̶̶ HK is a ̶̶ HK? 2.25 centimeters 4 centimeters 7.5 centimeters 9 centimeters 2. In △RST, SR < ST, and RT > ST. If m∠R = (2x + 10) ° and m∠T = (3x - 25) °, which is a possible value of x? 25 30 35 40 3. The vertex angle of an isosceles triangle measures (7a - 2) °, and one of the base angles measures (4a + 1) °. Which term best describes this triangle? Acute Equiang

ular Right Obtuse 4. The lengths of two sides of an acute triangle are 8 inches and 10 inches. Which of the following could be the length of the third side? 5 inches 6 inches 12 inches 13 inches 5. For the coordinates M (-1, 0), N (-2, 2), P (10, y), ̶̶̶ MN ǁ ̶̶ PQ. What is the value of y? and Q (4, 6), -18 -6 6 18 6. What is the area of an equilateral triangle that has a perimeter of 18 centimeters? 9 square centimeters 9 √  3 square centimeters 18 square centimeters 18 √  3 square centimeters 7. In △ABC and △DEF, ̶̶ AC ≅ ̶̶ DE, and ∠A ≅ ∠E. Which of the following would allow you to conclude by SAS that these triangles are congruent? ̶̶ AB ≅ ̶̶ AC ≅ ̶̶ BA ≅ ̶̶ CB ≅ ̶̶ DF ̶̶ EF ̶̶ FE ̶̶ DF 8. For the segment below, AB = 1 __ AC, and CD = 2BC. 2 Which expression is equal to the length of ̶̶̶ AD? 2AB + BC 2AC + AB 3AB 4BC 9. In △DEF, m∠D = 2 (m∠E + m∠F). Which term best describes △DEF? Acute Equiangular Right Obtuse 10. Which point of concurrency is always located inside the triangle? The centroid of an obtuse triangle The circumcenter of an obtuse triangle The circumcenter of a right triangle The orthocenter of a right triangle 374 374 Chapter 5 Properties and Attributes of Triangles �� If a diagram is not provided, draw your own. Use the given information to label the diagram. 11. The length of one leg of a right triangle is 3 times the length of the other, and the length of the hypotenuse is 10. What is the length of the longest leg? 3 3 √  10 √  10 12 √  5 12. Which statement is true by the Transitive Property of Congruence? If ∠A ≅ ∠T, then �

�T ≅ ∠A. If m∠L = m∠S, then ∠L ≅ ∠S. 5QR + 10 = 5 (QR + 2) ̶̶ ̶̶ ̶̶ EF, then DE ≅ DE and If ̶̶ BD ≅ ̶̶ BD ≅ ̶̶ EF. Gridded Response 13. P is the incenter of △JKL. The distance from P ̶̶ KL is 2y - 9. What is the distance from P to to ̶̶ JK? STANDARDIZED TEST PREP Short Response 17. In △RST, S is on the perpendicular bisector of m∠S = (4n + 16) °, and m∠R = (3n - 18) °. Find m∠R. Show your work and explain how you determined your answer. ̶̶ RT, 18. Given that ̶̶ BD ǁ ̶̶ AC and ̶̶ AB ≅ ̶̶ BD, explain why AC < DC. 19. Write an indirect proof that an acute triangle cannot contain a pair of complementary angles. Given: △XYZ is an acute triangle. Prove: △XYZ does not contain a pair of complementary angles. 20. Find the coordinates of the orthocenter of △JKL. Show your work and explain how you found your answer. 14. In a plane, r ǁ s, and s ⊥ t. How many right angles are formed by the lines r, s, and t? 15. What is the measure, in degrees, of ∠H? 16. The point T is in the interior of ∠XYZ. If m∠XYZ = (25x + 10) °, m∠XYT = 90°, and m∠TYZ = (9x) °, what is the value of x? Extended Response 21. Consider the statement “If a triangle is equiangular, then it is acute.” a. Write the converse, inverse, and contrapositive of this conditional statement. b. Write a biconditional statement from the conditional statement. c. Determine the truth value of the biconditional statement. If it is false, give a counterexample. d. Determine the truth value of each statement below. Give an example or count

erexample to justify your reasoning. “For any conditional, if the inverse and contrapositive are true, then the biconditional is true.” “For any conditional, if the inverse and converse are true, then the biconditional is true.” Cumulative Assessment, Chapters 1–5 375 375 �� Polygons and Quadrilaterals 6A Polygons and Parallelograms Lab Construct Regular Polygons 6-1 Properties and Attributes of Polygons Lab Explore Properties of Parallelograms 6-2 Properties of Parallelograms 6-3 Conditions for Parallelograms 6B Other Special Quadrilaterals 6-4 Properties of Special Parallelograms Lab Predict Conditions for Special Parallelograms 6-5 Conditions for Special Parallelograms Lab Explore Isosceles Trapezoids 6-6 Properties of Kites and Trapezoids KEYWORD: MG7 ChProj This tile mosaic showing the Alamo and surrounding buildings is on the Riverwalk in San Antonio. 376 376 Chapter 6 Vocabulary Match each term on the left with a definition on the right. 1. exterior angle A. lines that intersect to form right angles 2. parallel lines B. lines in the same plane that do not intersect 3. perpendicular lines C. two angles of a polygon that share a side 4. polygon 5. quadrilateral D. a closed plane figure formed by three or more segments that intersect only at their endpoints E. a four-sided polygon F. an angle formed by one side of a polygon and the extension of a consecutive side Triangle Sum Theorem Find the value of x. 6. 7. 8. 9. Parallel Lines and Transversals Find the measure of each numbered angle. 10. 11. 12. Special Right Triangles Find the value of x. Give the answer in simplest radical form. 13. 15. 14. 16. Conditional Statements Tell whether the given statement is true or false. Write the converse. Tell whether the converse is true or false. 17. If two angles form a linear pair, then they are supplementary. 18. If two angles are congruent, then they are right angles. 19. If a triangle is a scalene triangle, then it is an acute triangle. Polygons and Quadrilaterals 377 377 ��

�� Key Vocabulary/Vocabulario concave diagonal cóncavo diagonal isosceles trapezoid trapecio isósceles kite cometa parallelogram paralelogramo rectangle rectángulo regular polygon polígono regular rhombus square trapezoid rombo cuadrado trapecio Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word concave is made up of two parts: con and cave. Sketch a polygon that looks like it caves in. 2. If a triangle is isosceles, then it has two congruent legs. What do you think is a special property of an isosceles trapezoid? 3. A parallelogram has four sides. What do you think is a special property of the sides of a parallelogram? Geometry TEKS G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships 6-1 Geo. Lab ★ 6-2 Geo. Lab Les. 6-1 Les. 6-2 Les. 6-3 Les. 6-4 6-5 Tech. Lab 6-6 Tech. Lab Les. 6-5 Les. 6-6 ★ ★ ★ ★ ★ ★ G.2.B Geometric structure* make conjectures about... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ polygons... and determine the validity of the conjectures, choosing from a variety of approaches... G.3.B Geometric structure* construct and justify statements ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ about geometric figures and their properties G.3.E Geometric structure* use deductive reasoning to prove ★ ★ ★ ★ ★ a statement G.5.B Geometric patterns* use numeric and geometric ★ ★ patterns to make generalizations about geometric properties, including properties of polygons,... and angle relationships in polygons... G.7.A Dimensionality and the geometry of location* use one- and two-dimensional coordinate systems to represent... figures G.7.B Dimensionality and the geometry of location* use slopes and equations of lines to investigate geometric relationships... G.7.C Dimensionality and the geometry of location

*... use formulas involving length, slope, and midpoint ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ G.9.B Congruence and the geometry of size* formulate and ★ ★ ★ ★ test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models * Knowledge and skills are written out completely on pages TX28–TX35. 378 378 Chapter 6 Writing Strategy: Write a Convincing Argument Throughout this book, the exercises that require you to write an explanation or argument to support an idea. Your response to a Write About It exercise shows that you have a solid understanding of the mathematical concept. icon identifies To be effective, a written argument should contain • a clear statement of your mathematical claim. • evidence or reasoning that supports your claim. From Lesson 5-4 36. Write About It An isosceles triangle has two congruent sides. Does it also have two congruent midsegments? Explain. Step 1 Make a statement of your mathematical claim. Make a statement of your mathematical claim. Draw a sketch to investigate the properties of the midsegments of an isosceles triangle. You will find that the midsegments parallel to the legs of the isosceles triangle are congruent. Claim: The midsegments parallel to the legs of an isosceles triangle are congruent. Step 2 Give evidence to support your claim. Identify any properties or theorems that support your claim. In this case, the Triangle Midsegment Theorem states that the length of a midsegment of a triangle is 1 __ 2 the length of the parallel side. To clarify your argument, label your diagram and use it in your response. Step 3 Write a complete response. Write a complete response. Yes, the two midsegments parallel to the legs of an isosceles triangle ̶̶ ̶̶ are congruent. Suppose △ABC is isosceles with YZ XZ and are midsegments of △ABC. By the Triangle Midsegment Theorem, ̶̶ ̶̶ AC, AB = AC. So 1 __ 2 AB = 1 __ 2 AC XZ = 1 __ 2 AC and YZ = 1 __ 2 AB. Since AB ≅ by the Multiplication Property of Equality. By substitution, XZ = YZ, so ̶̶ AB ≅ ̶̶ XZ ≅ ̶

̶ AC. ̶̶ YZ. Try This Write a convincing argument. 1. Compare the circumcenter and the incenter of a triangle. 2. If you know the side lengths of a triangle, how do you determine which angle is the largest? Polygons and Quadrilaterals 379 379 6-1 Use with Lesson 6-1 Activity 1 Construct Regular Polygons In Chapter 4, you learned that an equilateral triangle is a triangle with three congruent sides. You also learned that an equilateral triangle is equiangular, meaning that all its angles are congruent. In this lab, you will construct polygons that are both equilateral and equiangular by inscribing them in circles. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures.... Also G.2.B, G.3.B, G.3.D, G.5.B, G.9.B 1 Construct circle P. Draw a diameter ̶̶ AC. 2 Construct the perpendicular bisector of of the bisector and the circle as B and D. ̶̶ AC. Label the intersections 3 Draw ̶̶ AB, ̶̶ BC, ̶̶ CD, and ̶̶ DA. The polygon ABCD is a regular quadrilateral. This means it is a four-sided polygon that has four congruent sides and four congruent angles. Try This 1. Describe a different method for constructing a regular quadrilateral. 2. The regular quadrilateral in Activity 1 is inscribed in the circle. What is the relationship between the circle and the regular quadrilateral? 3. A regular octagon is an eight-sided polygon that has eight congruent sides and eight congruent angles. Use angle bisectors to construct a regular octagon from a regular quadrilateral. Activity 2 1 Construct circle P. Draw a point A on the circle. 2 Use the same compass setting. Starting at A, draw arcs to mark off equal parts along the circle. Label the other points where the arcs intersect the circle as B, C, D, E, and F. 3 Draw ̶̶ BC, ̶̶ CD, ̶̶ AB, ̶̶ FA. The polygon ABCDEF is a regular hexagon. This means it is a six-sided polygon that has six congruent sides and six congruent angles

. ̶̶ EF, and ̶̶ DE, Try This 4. Justify the conclusion that ABCDEF is a regular hexagon. (Hint: Draw ̶̶ AD, ̶̶ BE, and ̶̶ CF. What types of triangles are formed?) diameters 5. A regular dodecagon is a 12-sided polygon that has 12 congruent sides and 12 congruent angles. Use the construction of a regular hexagon to construct a regular dodecagon. Explain your method. 380 380 Chapter 6 Polygons and Quadrilaterals �� Activity 3 1 Construct circle P. Draw a diameter ̶̶ AB. 2 Construct the perpendicular bisector of ̶̶ AB. Label one point where the bisector intersects the circle as point E. 3 Construct the midpoint of radius ̶̶ PB. Label it as point C. 4 Set your compass to the length CE. Place the compass point at C and draw an arc that intersects ̶̶ AB. Label the point of intersection D. 5 Set the compass to the length ED. Starting at E, draw arcs to mark off equal parts along the circle. Label the other points where the arcs intersect the circle as F, G, H, and J. 6 Draw ̶̶ EF, ̶̶ FG, ̶̶̶ GH, ̶̶ JE. The polygon EFGHJ is a regular pentagon. This means it is a five-sided polygon that has five congruent sides and five congruent angles. ̶̶ HJ, and Steps 1–3 Step 4 Step 5 Step 6 Try This 6. A regular decagon is a ten-sided polygon that has ten congruent sides and ten congruent angles. Use the construction of a regular pentagon to construct a regular decagon. Explain your method. 7. Measure each angle of the regular polygons in Activities 1–3 and complete the following table. REGULAR POLYGONS Number of Sides Measure of Each Angle Sum of Angle Measures 3 60° 180° 4 5 6 8. Make a Conjecture What is a general rule for finding the sum of the angle measures in a regular polygon with n sides? 9. Make a Conjecture What is a general rule for finding the measure of each angle in a regular polygon with n sides? 6-1 Geometry Lab 381 381 �� 6-1 Properties and Attributes

of Polygons TEKS G.5.B Geometric patterns: use … patterns to make generalizations about... properties of... and angle relationships in polygons.... Objectives Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Why learn this? The opening that lets light into a camera lens is created by an aperture, a set of blades whose edges may form a polygon. (See Example 5.) Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex Also G.2.B, G.3.B, G.4.A, G.5.A, G.7.A In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons. Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE is a pentagon. Number of Sides Name of Polygon 3 4 5 6 7 8 9 10 12 n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon E X A M P L E 1 Identifying Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. A B C polygon, pentagon not a polygon polygon, octagon Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1a. 1b. 1c. All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. 382 382 Chapter 6 Polygons and Quadrilaterals �� A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is

always convex. E X A M P L E 2 Classifying Polygons Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. A B C irregular, convex regular, convex irregular, concave Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 2a. 2b. To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Polygon Number of Sides Number of Triangles Triangle Quadrilateral Pentagon Hexagon n-gon Sum of Interior Angle Measures (1) 180° = 180° (2) 180° = 360° (3) 180° = 540° (4) 180° = 720° (n - 2) 180° In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n - 2) 180°. Theorem 6-1-1 Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180°. 6- 1 Properties and Attributes of Polygons 383 383 �� E X A M P L E 3 Finding Interior Angle Measures and Sums in Polygons A Find the sum of the interior angle measures of a convex octagon. (n - 2) 180° (8 - 2) 180° 1080° Polygon ∠ Sum Thm. An octagon has 8 sides, so substitute 8 for n. Simplify. B Find the measure of each interior angle of a regular nonagon. Step 1 Find the sum of the interior angle measures. (n - 2) 180° (9 - 2) 180° = 1260° Polygon ∠ Sum Thm. Substitute 9 for n and simplify. Step 2 Find the measure of one interior angle. 1260° _ 9 = 140° The int.  are ≅, so divide by 9. C Find the measure of each interior angle of quadrilateral

PQRS. (4 - 2) 180° = 360° Polygon ∠ Sum Thm. Polygon ∠ Sum Thm. m∠P + m∠Q + m∠R + m∠S = 360° c + 3c + c + 3c = 360 8c = 360 c = 45 Substitute. Combine like terms. Divide both sides by 8. m∠P = m∠R = 45° m∠Q = m∠S = 3 (45°) = 135° 3a. Find the sum of the interior angle measures of a convex 15-gon. 3b. Find the measure of each interior angle of a regular decagon. In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°. An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Theorem 6-1-2 Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. E X A M P L E 4 Finding Exterior Angle Measures in Polygons A Find the measure of each exterior angle of a regular hexagon. A hexagon has 6 sides and 6 vertices. sum of ext.  = 360° measure of one ext. ∠ = 360° _ = 60° 6 Polygon Ext. ∠ Sum Thm. A regular hexagon has 6 ≅ ext. , so divide the sum by 6. The measure of each exterior angle of a regular hexagon is 60°. 384 384 Chapter 6 Polygons and Quadrilaterals �� B Find the value of a in polygon RSTUV. 7a° + 2a° + 3a° + 6a° + 2a° = 360° Polygon Ext. ∠ Sum Thm. 20a = 360 a = 18 Combine like terms. Divide both sides by 20. 4a. Find the measure of each exterior angle of a regular dodecagon. 4b. Find the value of r in polygon JKLM. E X A M P L E 5 Photography Application The aperture of the camera is formed by ten blades. The blades overlap to form a