jacobcd52/hs_math_cleaned · Datasets at Hugging Face

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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: ...
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 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° Su...
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 a...
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 ...
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 an...
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 �� KEYWOR...
 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 equ...
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, w...
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 ann...
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 ...
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 i...
, 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... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★...
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...
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 + ...
.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 ̶̶ ̶̶ Si...
�� 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). ̶...
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 =...
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) °, fin...
�� ̶̶ 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 ̶...
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 par...
(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 ...
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 ��...
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...
�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...
-6560�� 5-2 Exercises Exercises KEYWORD: MG...
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, Gaspa...
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: ...
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 th...
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 concur...
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 verti...
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)....
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...
(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 w...
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?...
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∠J...
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...
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 _...
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 T...
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 ...
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 fo...
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 coordina...
. ̶̶̶ 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% ac...
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...
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...
-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,...
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 ...
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 opp...
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...
+ 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 ...
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 ...
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 a...
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 di...
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 Additio...
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 ...
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...
∠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 ̶̶ Q...
< 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 to...
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 GUI...
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 ...
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 stat...
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 mapGeometr...
. √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 I...
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 sta...
�� (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...
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. Tel...
, 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 ...
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 meas...
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 answ...
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 p...
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 ...
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 amon...
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 ...
= 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 ...
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 ��...
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 a...
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....
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 a...
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 Re...
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...
........ 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 Pythagore...
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 conclu...
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 ort...
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...
> 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....
. 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 th...
. 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 obtu...
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...
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 ver...
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 d...
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. ...
�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 bi...
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 ��...
�� 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 cuadrad...
... 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...
̶ 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, y...
. ̶̶ 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 cons...
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 ...
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...
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 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: ...

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 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° Su...

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 a...

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 ...

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 an...

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 �� KEYWOR...

 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 equ...

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, w...

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 ann...

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 ...

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 i...

, 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... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★...

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...

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 + ...

.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 ̶̶ ̶̶ Si...

�� 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). ̶...

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 =...

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) °, fin...

�� ̶̶ 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 ̶...

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 par...

(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 ...

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 ��...

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...

�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...

-6560�� 5-2 Exercises Exercises KEYWORD: MG...

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, Gaspa...

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: ...

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 th...

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 concur...

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 verti...

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)....

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...

(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 w...

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?...

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∠J...

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...

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 _...

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 T...

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 ...

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 fo...

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 coordina...

. ̶̶̶ 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% ac...

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...

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...

-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,...

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 ...

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 opp...

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...

+ 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 ...

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 ...

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 a...

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 di...

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 Additio...

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 ...

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...

∠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 ̶̶ Q...

< 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 to...

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 GUI...

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 ...

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 stat...

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 mapGeometr...

. √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 I...

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 sta...

�� (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...

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. Tel...

, 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 ...

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 meas...

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 answ...

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 p...

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 ...

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 amon...

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 ...

= 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 ...

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 ��...

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 a...

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....

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 a...

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 Re...

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...

........ 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 Pythagore...

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 conclu...

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 ort...

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...

> 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....

. 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 th...

. 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 obtu...

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...

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 ver...

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 d...

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. ...

�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 bi...

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 ��...

�� 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 cuadrad...

*... 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...

̶ 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, y...

. ̶̶ 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 cons...

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 ...

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...

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 ...