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�� MN ⊥ PQ for M (2, 1), N (-3, 0), P (x, 4), and Q (3, y). Find a set of possible values for x and y. SPIRAL REVIEW Find the x- and y-intercepts of the line that contains each pair of points. (Previous course) 34. (-5, 0) and (0, -5) 35. (0, 1) and (2, -7) 36. (1, -3) and (3, 3) Use the given paragraph proof t... |
x – 4, y = –3x – 4, and y = 3x + 1. Which lines appear to be parallel? What do you notice about the slopes of the parallel lines? 2 Graph y = 2x. Experiment with other equations to find a line that appears parallel to y = 2x. If necessary, graph y = 2x on graph paper and construct a parallel line. What is the slope of ... |
Do the lines still appear perpendicular? Describe your results. 6. Try changing the y-intercepts of one of the perpendicular lines. Does this change whether the lines appear to be perpendicular? 3- 6 Technology Lab 189 189 3-6 Lines in the Coordinate Plane TEKS G.7.B Dimensionality and the geometry of location: use...... |
the linex - x 1 ) m = (x - x 1 ) m (x - x 1 ) = ( Slope formula Substitute (x, y) for ( x 2, y 2 ). Multiply both sides by (x - x 1 ). Simplify. y - y 1 = m (x - x 1 ) Sym. Prop. of = 190 190 Chapter 3 Parallel and Perpendicular Lines E X A M P L E 1 Writing Equations of Lines Write the equation of each line in the gi... |
x - 1) The equation is given in point-slope form, with a slope of -2 = -2 ___ 1 through the point (1, -3). Plot the point (1, -3) and then rise -2 and run 1 to find another point. Draw the line containing the two points. 3- 6 Lines in the Coordinate Plane 191 191 ��������������������������������������������������������... |
mine whether the lines 3x + 5y = 2 and 3x + 6 = -5y are parallel, intersect, or coincide. 192 192 Chapter 3 Parallel and Perpendicular Lines ��������������������� E X A M P L E 4 Problem-Solving Application Audrey is trying to decide between two health club plans. After how many months would both plans’ total costs be ... |
. Vocabulary How can you recognize the slope-intercept form of an equation Write the equation of each line in the given form. p. 191 2. the line through (4, 7) and (-2, 1) in slope-intercept form 3. the line through (-4, 2) with slope 3 _ 4 4. the line with x-intercept 4 and y-intercept -2 in slope-intercept form in po... |
2y = 5x - 4 22. 7x + 2y = 10, 3y = 4x - 5 23. Business Chris is comparing two sales positions that he has been offered. The first pays a weekly salary of $375 plus a 20% commission. The second pays a weekly salary of $325 plus a 25% commission. How much must he make in sales per week for the two jobs to pay the same? ... |
5, 1) 43. G (3, 4), H (-3, 4), J (1, -2) 44. K (-2, 4), L (2, 1), M (1, 8) In 2004, the world’s largest pizza was baked in Italy. The diameter of the pizza was 5.19 m (about 17 ft) and it weighed 124 kg (about 273 lb). 45. Food A restaurant charges $8 for a large cheese pizza plus $1.50 per topping. Another restaurant ... |
. On the same graph you made for part a, graph the line d = 300. What does the intersection of the two lines represent? c. Use the graph to estimate the number of seconds it takes the car to travel 300 ft. 54. Prove the slope-intercept form of a line, given the point-slope form. Given: The equation of the line through ... |
, find a possible equation that describes the line that contains the hypotenuse. 64. Find the equations of three lines that form a triangle with a hypotenuse of 13 units. 65. Multi-Step Are the points (-2, -4), (5, -2) and (2, -3) collinear? Explain the method you used to determine your answer. 66. For the line y = x +... |
passes through all the points. x 30 60 90 120 150 y = 180 - x 150 120 90 60 30 Step 3 Choose two points from the line, such as (30, 150) and (120, 60). m = Use them to find the slope = 60 - 150 _ 120 - 30 = -90 _ 90 = -1 Slope formula Substitute (30, 150) for ( x 1, y 1 ) and (120, 60) for ( x 2, y 2 ). Simplify. Step... |
6, 62) seems to be closest to all the points. Draw this line. Use the points (2, 36) and (6, 62) to find the slope of the line = 62 - 36 _ _ 6 - 2 = 6.5 Substitute (2, 36) for ( x 1, y 1 ) and (6, 62) for ( x 2, y 2 ). Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - ... |
of the critical distance for a yellow light on Porter Street and then graph the line. Does this line intersect the line for Lincoln Road? If so, where? Is the line for Porter Street steeper or flatter than the line for Lincoln Road? Explain how you know. 200 200 Chapter 3 Parallel and Perpendicular Lines �������������... |
. ST and VW for S (0, 3), T (0, 7), V (2, 3), and W (5, 3) 3-6 Lines in the Coordinate Plane Write the equation of each line in the given form. 14. the line through (3, 8) and (-3, 4) in slope-intercept form 15. the line through (-5, 4) with slope 2 _ 3 16. the line with y-intercept 2 through the point (4, 1)... |
............ 147 run........................ 182 Complete the sentences below with vocabulary words from the list above. 1. Angles on opposite sides of a transversal and between the lines it intersects are?. ̶̶̶̶ 2. Lines that are in different planes are?. ̶̶̶̶ 3. A(n) 4. The? is a line that intersects two coplanar lin... |
.E, G.9.A By the Same-Side Interior Angles Theorem, (6x + 10) + (4x + 20) = 180. 15. m∠KLM x = 15 Solve for x. Substitute the value for x into the expression for m∠TUV. m∠TUV = 4 (15) + 20 = 80° ■ m∠ABC 16. m∠DEF 17. m∠QRS By the Corresponding Angles Postulate, 8x + 28 = 10x + 4. x = 12 Solve for x. Substitute the valu... |
15, x = 9 m∠1 = 60°, and m∠5 = 60°. So ∠1 ≅ ∠5. p ǁ q by the Converse of the Alternate Exterior Angles Theorem. 3-4 Perpendicular Lines (pp. 172–178) TEKS G.1.A, G.2.A, G.3.C, G.3.E, G.9.A EXERCISES 22. Name the shortest segment from point K to ̶̶ LN. 23. Write and solve an inequality for x. 24. Given: Prove: ̶̶ AD ǁ ... |
(3, -1), and D (4, -33 - (-1) -3 - (-1) = -2 _ _ 4 - 3 1 The slopes are opposite reciprocals, so the slope of CD = = -2 Use slopes to determine if the lines are parallel, perpendicular, or neither. 27. EF and GH for E (8, 2), F (-3, 4), G (6, 1), and H (-4, 3) 28. JK and LM for J (4, 3), K (-4, -2), L... |
the lines are parallel, intersect, or coincide. 33. -3x + 2y = 5, 6x - 4y = 8 34. y = 4x - 3, 5x + 2y = 1 35. y = 2x + 1, 2x - y = -1 Study Guide Review 205 205 ��������������������������������� Identify each of the following. 1. a pair of parallel planes 2. a pair of parallel segments 3. a pair of skew segments Find ... |
US ON ACT When you take the ACT Mathematics Test, you receive a separate subscore for each of the following areas: • Pre-Algebra/Elementary Algebra, • Intermediate Algebra/Coordinate Geometry, and • Plane Geometry/Trigonometry. You may want to time yourself as you take this practice test. It should take you about 5 min... |
Exam Practice 207 207 ����������� Any Question Type: Interpret Coordinate Graphs When test items refer to a coordinate plane, it is important to interpret the coordinate graphs correctly. It is also important to understand the relationship between an equation and its graph. Multiple Choice Which statement best describ... |
y-intercept of the line through (5, 2) that is parallel to the line x - 4y = 8? y = 3x - 10 y = -3x + 14. Graph the line represented by 3x - y = 7. What is its slope? 8. Is the slope of a line perpendicular to the line represented by 3x - y = 7 positive or negative? Based on your answer, can you eliminate any answer c... |
-2x + = 2x + 10 7. Given the points R (-5, 3), S (-5, 4), T (-3, 4), and U (-3, 1), which line is perpendicular to TU? RS RT ST SU 8. Which of following is true if XY and UV are skew? XY and UV are coplanar. X, Y, and U are noncollinear. XY ǁ UV XY ⊥ UV ... |
= n + 7. 13. Which condition guarantees that r || s? STANDARDIZED TEST PREP Short Response 21. Given ℓ ǁ m with transversal t, explain why ∠1 and ∠8 are supplementary. ∠1 ≅ ∠2 ∠2 ≅ ∠7 ∠2 ≅ ∠3 ∠1 ≅ ∠4 22. Read the following conditional statement. If two angles are vertical angles, then they are congruent. a. Write the ... |
. Nelson wants to put a chain-link fence around 3 sides of a square-shaped lawn. Chain-link fencing is sold in sections that are each 6 feet wide. If Ms. Nelson’s lawn has an area of 3600 square feet, how many sections of fencing will she need? 20. What is the next number in this pattern? 67, 76, 83, 88,… 24. A car pas... |
measure each angle. 6. 7. Use a protractor to draw an angle with each of the following measures. 8. 20° 10. 105° 9. 63° 11. 158° Solve Equations with Fractions x + 7 = 25 Solve. 12. 9_ 2 14. x - 1_ 5 = 12_ 5 13. 3x - 2_ 3 = 4_ 3 15. 2y = 5y - 21_ 2 Connect Words and Algebra Write an equation for each statement. 16. Ta... |
conjectures about angles, lines, polygons … and determine validity of the conjectures, choosing from a variety of approaches … 4-2 Geo. Lab Les. 4-1 ★ Les. 4-2 Les. 4-3 ★ 4-4 Geo. Lab 4-5 Tech. Lab Les. 4-4 Les. 4-5 Les. 4-6 Les. 4-7 Les. 4-8 Ext.5.A Geometric patterns* use … geometric patterns ★ to develop algebraic ... |
3. Segment XY is perpendicular to line BC. 4. If not p, then not q. Translate the symbols into words. 5. m∠FGH = m∠VWX 6. ZA ǁ TU 7. ∼p → q 8. ST bisects ∠TSU. Triangle Congruence 215 215 4-1 Classifying Triangles TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, ... |
s Equilateral Triangle Isosceles Triangle Scalene Triangle Three congruent sides At least two congruent sides No congruent sides E X A M P L E 2 Classifying Triangles by Side Lengths Classify each triangle by its side lengths. When you look at a figure, you cannot assume segments are congruent based on their appearance... |
12 in. To find the number of triangles that can be made from 100 inches. of steel, divide 100 by the amount of steel needed for one triangle. 100 ÷ 12 = 8 1 _ 3 triangles There is not enough steel to complete a ninth triangle. So the manufacturer can make 8 triangles from a 100 in. piece of steel. Each measure is the ... |
AKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Classify each triangle by its angle measures. 12. △BEA 13. △DBC 14. △ABC Classify each triangle by its side lengths. 15. △PST 16. △RSP 17. △RPT Multi-Step Find the side lengths of each triangle. 18. 19. ���� ������ 20. Draw a trian... |
Twenty-second Street. The Fifth Avenue side is 1 ft shorter than twice the East Twenty-second Street side. The East Twenty-second Street side is 8 ft shorter than half the Broadway side. The Broadway side is 190 ft. a. Find the two unknown side lengths. b. Classify the triangle by its side lengths. 34. Critical Thinki... |
vertices with coordinates (0, 0), (a, 0), and (0, a), where a ≠ 0. Classify the triangle in two different ways. Explain your answer. 46. Write a two-column proof. Given: △ABC is equiangular. EF ǁ AC Prove: △EFB is equiangular. 47. Two sides of an equilateral triangle measure (y + 10) units and ( y 2 - 2) units. If the... |
Place the patty paper on top of the triangle you drew. Align the papers so that coincide. Trace ∠B. Rotate the triangle and trace ∠C adjacent to ∠B. Rotate the triangle again and trace ∠A adjacent to ∠C. The diagram shows your final step. Try This 1. What do you notice about the three angles of the triangle that you t... |
������������������������������������������������� E X A M P L E 1 Surveying Application The map of France commonly used in the 1600s was significantly revised as a result of a triangulation land survey. The diagram shows part of the survey map. Use the diagram to find the indicated angle measures. A m∠NKM m∠KMN + m∠MNK... |
2-2 and 4-2-3 in Exercises 24 and 25. 224 224 Chapter 4 Triangle Congruence 70°104°88°48°������ E X A M P L E 2 Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures 22.9°. What is the measure of the other acute angle? Let the acute angles be ∠M and ∠N, with m∠M = 22.9°. m∠M + m... |
Find m∠ACD. 4-2 Angle Relationships in Triangles 225 225 ������������������������������������������������������������������������� Theorem 4-2-5 Third Angles Theorem THEOREM HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.... |
E of △DEF. What are its remote interior angles? 3. What do you call segments, rays, or lines that are added to a given diagram. 224 Astronomy Use the following information for Exercises 4 and 5. An asterism is a group of stars that is easier to recognize than a constellation. One popular asterism is the Summer Triangl... |
angle measure. 19. m∠XYZ 20. m∠C 21. m∠N and m∠P 22. m∠Q and m∠S 23. Multi-Step The measures of the angles of a triangle are in the ratio 1 : 4 : 7. What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the angle measures.) 24. Complete the proof of Corollary 4-2-2. Given: △DEF with right ∠F Prove: ∠... |
What is the measure of any exterior angle of an equiangular triangle? What is the sum of the exterior angle measures? 34. Find m∠SRQ, given that ∠P ≅ ∠U, ∠Q ≅ ∠T, and m∠RST = 37.5°. 35. Multi-Step In a right triangle, one acute angle measure is 4 times the other acute angle measure. What is the measure of the smaller ... |
41. What is the value of x? 19 52 42. Find the value of s. 23 28 57 71 34 56 43. ∠A and ∠B are the remote interior angles of ∠BCD in △ABC. Which of these equations must be true? m∠A - 180° = m∠B m∠A = 90° - m∠B m∠BCD = m∠BCA - m∠A m∠B = m∠BCD - m∠A 44. Extended Response The measures of the angles in a triangle are in ... |
56. △ABD 57. What if…? If CA = 8, What is the effect on the classification of △ACD? 230 230 Chapter 4 Triangle Congruence ������������������������������������������������������ 4-3 Congruent Triangles TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.B Objec... |
XYZ represent the triangles of the space station’s support structure. If △RST ≅ △XYZ, identify all pairs of congruent corresponding parts. Angles: ∠R ≅ ∠X, ∠S ≅ ∠Y, ∠T ≅ ∠Z ̶̶ ST ≅ Sides: ̶̶ RT ≅ ̶̶ RS ≅ ̶̶ XY, ̶̶ YZ, ̶̶ XZ 1. If polygon LMNP ≅ polygon EFGH, identify all pairs of corresponding congruent parts. 4-3 Cong... |
Given: ∠P and ∠M are right angles. R is the midpoint of ̶̶ ̶̶ NR PQ ≅ ̶̶ PM. ̶̶̶ MN, ̶̶ QR ≅ Prove: △PQR ≅ △MNR Proof: Statements Reasons 1. ∠P and ∠M are rt. 1. Given 2. ∠P ≅ ∠M 3. ∠PRQ ≅ ∠MRN 4. ∠Q ≅ ∠N ̶̶̶ PM. 6. 5. R is the mdpt. of ̶̶̶ MR ̶̶̶ MN ; ̶̶ PR ≅ ̶̶ PQ ≅ ̶̶ QR ≅ 7. ̶̶ NR 2. Rt. ∠ ≅ Thm. 3. Vert. Thm.... |
�LKM, ̶̶ MK ̶̶ JK ⊥ ̶̶ JK ≅ ̶̶ KL, ̶̶ ML, ̶̶ JK ⊥ ̶̶ KL, 1. Statements ̶̶̶ ML ⊥ ̶̶ KL Reasons 1. Given 2. ∠JKL and ∠MLK are rt. . 2. Def. of ⊥ lines 3. ∠JKL ≅ ∠MLK 4. ∠KLJ ≅ ∠LKM 5. ∠KJL ≅ ∠LMK ̶̶ JK ≅ ̶̶ KL ≅ ̶̶̶ ML, ̶̶ LK 6. 7. ̶̶ JL ≅ ̶̶̶ MK 8. △JKL ≅ △MLK 3. Rt. ∠ ≅ Thm. 4. Given 5. Third Thm. 6. Given 7. Reflex... |
. Identify the congruent corresponding parts. ̶̶ RS ≅ ̶̶ TS ≅ 3. 6.? ̶̶̶̶? ̶̶̶̶ ̶̶ LN ≅ 4. 7. ∠L ≅? ̶̶̶̶? ̶̶̶̶ 5. ∠S ≅ 8. ∠N ≅? ̶̶̶̶? ̶̶̶̶ Given: △FGH ≅ △JKL. Find each value. p. 232 9. KL 10. 232. 233 11. Given: E is the midpoint of ̶̶ AB ≅ ̶̶ CD, ̶̶ AB ǁ ̶̶ CD ̶̶ AC and ̶̶ BD. Prove: △ABE ≅ △CDE Proof: Statements Rea... |
. ∠F ≅? ̶̶̶̶? ̶̶̶̶ 14. ̶̶ KN ≅ 16. ∠L ≅? ̶̶̶̶? ̶̶̶̶ For See Exercises Example 13–16 17–18 19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 Given: △ABD ≅ △CBD. Find each value. 17. m∠C 18. y 19. Given: ̶̶̶ MP bisects ∠NMR. P is the midpoint of ̶̶ NR. ̶̶̶ MR, ∠N ≅ ∠R ̶̶̶ MN ≅ Prove: △M... |
different congruence statements. ̶̶ GR ≅ ̶̶ SR ≅ ̶̶ KP, 22. The two polygons in the diagram are congruent. Complete the following congruence statement for the polygons. polygon R? ≅ polygon V ̶̶̶̶? ̶̶̶̶ Write and solve an equation for each of the following. 23. △ABC ≅ △DEF. AB = 2x - 10, and DE = x + 20. Find the valu... |
HKL is congruent to △YWX? Explain. 31. Which congruence statement correctly indicates that the two given triangles are congruent? △ABC ≅ △DEF △ABC ≅ △FED △ABC ≅ △EFD △ABC ≅ △FDE 32. △MNP ≅ △RST. What are the values of x and y? x = 26, y = 21 1 _ 3 x = 27, y = 20 x = 25, y = 20 2 _ 3, y = 16 2 _ x = 30 1 _ 3 3 33. △ABC ... |
COA = 140° Find each angle measure. (Lesson 4-2) 43. ∠Q 44. ∠P 45. ∠QRS KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra 1 and 2, Geometry, Precalculus Q: What kind of degree or certification will you receive? A: I will receive an associate’s degree in applied science. Then I will take a... |
and by its angle measures. ̶̶ ̶̶ DB bisects ∠ABC and ∠ADC. DE bisects ∠ADB. Find the measures of the angles in △EDB. Explain how you found the measures. ̶̶ 3. Given that DB bisects ∠ABC and ̶̶ ∠EDF, DE ≅ prove that △EDB ≅ △ FDB. ̶̶ BF, and ̶̶ BE ≅ ̶̶ DF, � � � � � � 238 238 Chapter 4 Triangle Congruence SECTION 4A Qui... |
�CDA ̶̶ AC ⊥ ̶̶ CD, ̶̶ DB ⊥ ̶̶ AB 3. 4. ∠ACD and ∠DBA are rt. 5. e.? ̶̶̶̶̶? ̶̶̶̶̶ ̶̶ CD, 6. f. ̶̶ AB ≅ 7. ̶̶ AC ≅ ̶̶ BD 8. h.? ̶̶̶̶̶ 9. △ACD ≅ △DBA 1. a. 2. b. 3. c. 4. d.? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶ 5. Rt. ∠ ≅ Thm. 6. Third Thm. 7. g.? ̶̶̶̶̶ 8. Reflex Prop. of ≅ 9. i.? ̶̶̶̶̶ Ready to Go On? 239 239 ���������������... |
to make two triangles that have the same side lengths but that are not congruent? Why or why not? 3. How does your answer to Problem 2 provide a shortcut for proving triangles congruent? 4. Complete the following conjecture based on your results. Two triangles are congruent if?. ̶̶̶̶̶̶̶̶̶̶̶̶̶ 240 240 Chapter 4 Triangl... |
congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triang... |
why △KPN ≅ △LPM. It is given that ̶̶ NP ≅ and that By the Vertical Angles Theorem, ∠KPN ≅ ∠LPM. Therefore △KPN ≅ △LPM by SAS. ̶̶ KP ≅ ̶̶̶ MP. ̶̶ LP 2. Use SAS to explain why △ABC ≅ △DBC. The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angle, you can construct... |
≅ △CDB Proving Triangles Congruent ̶̶ EG ≅ ̶̶ Given: ℓ ǁ m, HF Prove: △EGF ≅ △HFG Proof: Statements Reasons ̶̶ EG ≅ ̶̶ HF 1. 2. ℓ ǁ m 3. ∠EGF ≅ ∠HFG ̶̶ FG ≅ ̶̶ FG 4. 1. Given 2. Given 3. Alt. Int. Thm. 4. Reflex Prop. of ≅ 5. △EGF ≅ △HFG 5. SAS Steps 1, 3, 4 4. Given: Prove: △RQP ≅ △SQP QP bisects ∠RQS. ̶̶ QR ≅ ... |
18 4-4 Triangle Congruence: SSS and SAS 245 245 ���������������������������������������������������������������������������������������������������. Given: ̶̶ JK ≅ ̶̶̶ ML, ∠JKL ≅ ∠MLK p. 244 Prove: △JKL ≅ △MLK Proof: Statements Reasons ̶̶ JK ≅ ̶̶̶ ML 1. 2. b. ̶̶ KL ≅? ̶̶̶̶ ̶̶ LK 3. 4. △JKL ≅ △MLK 1. a.? ̶̶̶̶ 2. Given ... |
. 3. Given 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶ 6. Reflex. Prop. of ≅ 7. g.? ̶̶̶̶ ����������������������������������������������������������������������������������������������������������������� Which postulate, if any, can be used to prove the triangles congruent? 14. 16. 15. 17. 18. Explain what additional information, if any, y... |
�ZVY ≅ △WYV 7. Reflex. Prop. of ≅ 8. g.? ̶̶̶̶ 22. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The diagram shows two triangular trusses that were built for the roof of a doghouse. a. You can use a protractor to check that ∠A and ∠D are right angles. Explain how you could make just two additio... |
�JKL using SAS. ̶̶ ̶̶ JL. What additional JK and ̶̶ FH ≅ ̶̶ He knows that FG ≅ piece of information does he need? ∠H ≅ ∠L ∠F ≅ ∠G ∠F ≅ ∠J ∠G ≅ ∠K 31. What must the value of x be in order to prove that △EFG ≅ △EHG by SSS? 1.5 4.25 4.67 5.5 248 248 Chapter 4 Triangle Congruence �������������������������������������������... |
EG 44. m∠FGH Using Technology Use geometry software to complete the following. 1. Draw a triangle and label the vertices A, B, and C. Draw a point and label it D. Mark a vector from A to B and translate D by the marked vector. Label the image E. Draw DE. Mark ∠BAC and rotate DE about D by the marked angle. Ma... |
measure of ∠DGA in Step 2 will always be the same? 4. In Step 3 of the activity, the angle measures in △ADG stayed the same as the size of the triangle changed. Does Angle-Angle-Angle, like Side-Side-Side, make only one triangle? Explain. 5. Repeat Step 4 of the activity but measure the length of ̶̶ AG instead of ̶̶ A... |
, and HL. Vocabulary included side Why use this? Bearings are used to convey direction, helping people find their way to specific locations. Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar course. Directions are given by bearings, which are based on comp... |
checkpoints. 1. What if...? If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. E X A M P L E 2 Applying ASA Congruence Determine if you can use ASA to prove △UVX ≅ △WVX. Explain. ∠UXV ≅ ∠WXV as given. Since ∠WVX is a right angle that forms a li... |
̶̶ HJ ≅ 3. 2. Third Thm. 3. Given 4. △GHJ ≅ △KLM 4. ASA Steps 1, 3, and 2 E X A M P L E 3 Using AAS to Prove Triangles Congruent ̶̶ AB ǁ Use AAS to prove the triangles congruent. ̶̶ BC ≅ Given: Prove: △ABC ≅ △EDC Proof: ̶̶ ED, ̶̶ DC 3. Use AAS to prove the triangles congruent. ̶̶ JL bisects ∠KLM. ∠K ≅ ∠M Given: Prov... |
�YXZ This conclusion cannot be proved by HL. According to the diagram, △VWZ and △YXZ are right triangles, ̶̶̶ WZ and is congruent to hypotenuse ̶̶ XY. You do not know that hypotenuse ̶̶ XZ. ̶̶̶ WV ≅ 4. Determine if you can use the HL Congruence Theorem to prove △ABC ≅ △DCB. If not, tell what else you need to know. THIN... |
̶̶ QR ǁ ̶̶ SP Prove: △QPS ≅ △SRQ Proof. 255 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 7. △ABC and △CDA 8. △XYV and △ZYV 256 256 Chapter 4 Triangle Congruence CBA115 ft ge07sec_04l05003aa�������������������������������������������������... |
ence statement. Identify the transformation that moves one triangle to the position of the other triangle. 16. 17. 18. Critical Thinking Side-Side-Angle (SSA) cannot be used to prove two triangles congruent. Draw a diagram that shows why this is true. 4-5 Triangle Congruence: ASA, AAS, and HL 257 257 ������������������... |
̶̶̶ LM, ∠JMK ≅ ∠LMK ̶̶ JL, 24. Write About It The legs of both right △DEF and right △RST are 3 cm and 4 cm. They each have a hypotenuse 5 cm in length. Describe two different ways you could prove that △DEF ≅ △RST. 25. Construction Use the method for constructing perpendicular lines to construct a right triangle. 26. Wh... |
lie in intersecting planes. From the given angle measures, can you conclude that △VSU ≅ △VTU? Explain. m∠VUS = (7y - 2) ° m∠VUT = (5 1 _ m∠USV = 5 2 _ y ° 3 m∠SVU = (3y - 6) ° m∠TVU = 2x ° ) x - 1 _ 2 2 m∠UTV = (4x + 8) ° ° � � � � 32. Given: △ABC is equilateral. C is the midpoint of ̶̶ DE. ∠DAC and ∠EBC are congruent... |
congruent. E X A M P L E 1 Engineering Application SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. To design a bridge across a canyon, you need to find the distance from A to B. Locate points C, D, and E as shown in... |
and 4 6. ∠EDG ≅ ∠FGD ̶̶ GF ̶̶ ED ǁ 7. 6. CPCTC 7. Converse of Alt. Int. Thm. 3. Given: J is the midpoint of Prove: ̶̶ KL ǁ ̶̶̶ MN ̶̶̶ KM and ̶̶ NL. You can also use CPCTC when triangles are on a coordinate plane. You use the Distance Formula to find the lengths of the sides of each triangle. Then, after showing that... |
4) ) 16 + 9 = √ 25 = 5 = √ 2 EF = √ (-1 - 0) 2 + (-2 - 2) 2 = √ 1 + 16 = √ 17 DF = 2 √ + (-2 - (-1) ) (-1 - (-4) ) 9 + 1 = √ 10 = √ 2 ̶̶ AB ≅ ̶̶ BC ≅ So and ∠ABC ≅ ∠DEF by CPCTC. ̶̶ EF, and ̶̶ DE, ̶̶ AC ≅ ̶̶ DF. Therefore △ABC ≅ △DEF by SSS, 4. Given: J (-1, -2), K (2, -1), L (-... |
= 6.3 m, what is AB? KEYWORD: MG7 4-6 KEYWORD: MG7 Parent � � � � � �. 260 3. Given: X is the midpoint of ̶̶ ST. ̶̶ RX ⊥ ̶̶ ST Prove: ̶̶ RS ≅ ̶̶ RT Proof: 262 262 Chapter 4 Triangle Congruence ������������������������������������������������������������������������������������������������������������������������������... |
measures the distance from C to B. Then he locates point D the same distance east of C. If DE = 420 ft, what is AB? 8. 8. Given: M is the midpoint of ̶̶ PQ and ̶̶ ̶̶ PS QR ≅ Prove: ̶̶ RS. 9. Given: ̶̶̶ WX ≅ ̶̶ XY ≅ ̶̶ YZ ≅ ̶̶̶ ZW Prove: ∠W ≅ ∠Y 10. Given: G is the midpoint of ̶̶ FH. 11. Given: ̶̶̶ LM bisects ∠JLK. Pro... |
c. What is the length of ̶̶ BD and ̶̶ BD ≅ ̶̶ BC to the nearest tenth? ������ � ������ ������ �� � � Multi-Step Find the value of x. 17. 18. ����������������� Use the diagram for Exercises 19–21. 19. Given: PS = RQ, m∠1 = m∠4 Prove: m∠3 = m∠2 20. Given: m∠1 = m∠2, m∠3 = m∠4 Prove: PS = RS 21. Given: PS = RQ, PQ = RS P... |
, 4), (0, 6) CHALLENGE AND EXTEND 29. All of the edges of a cube are congruent. All of the angles on each face of a cube are right angles. Use CPCTC to explain why any two diagonals on the faces of a cube (for example, must be congruent. ̶̶ AC and ̶̶ AF ) 30. Given: ̶̶ JK ≅ ̶̶̶ ML, ̶̶ JM ≅ ̶̶ KL Prove: ∠J ≅ ∠L (Hint: D... |
� △ECD. Explain. (Lesson 4-5) 4-6 Triangle Congruence: CPCTC 265 265 ���������������������������������� Quadratic Equations Algebra A quadratic equation is an equation that can be written in the form a x 2 + bx + c = 0. See Skills Bank page S66 Example Given: △ABC is isosceles with ̶̶ AB ≅ ̶̶ AC. Solve for x. Step 1 Se... |
���������������������������������� 4-7 Introduction to Coordinate Proof TEK G.2.B Geometric structure: make conjectures about... polygons … and determine validity of the conjectures. Also G.3.B, G.9.B, G.10.B Objectives Position figures in the coordinate plane for use in coordinate proofs. Prove geometric concepts by u... |
C (4, 0). D is the ̶̶ AC. midpoint of Prove: The area of △DBC is one half the area of △ABC. Proof: △ABC is a right triangle with height AB and base BC. area of △ABC = 1 __ 2 bh = 1 __ 2 (4) (6) = 12 square units, 6 + 0 ____ 2 By the Midpoint Formula, the coordinates of D = ( 0 + 4 ____ 2 of △DBC, and the base is 4 uni... |
. The coordinates of A are (0, 2 j), the coordinates of B are (0, 0), and the coordinates of C are (2n, 0). Since you will use the Midpoint Formula to find the coordinates of D, use multiples of 2 for the leg lengths. Step 2 Position the figure in the coordinate plane. Step 3 Write a coordinate proof. Proof: △ABC is a ... |
plane. p. 267 2. a rectangle with a length of 4 units and width of 1 unit 3. a right triangle with leg lengths of 1 unit and 3 units Write a proof using coordinate geometry. p. 268 4. Given: Right △PQR has coordinates P (0, 6), Q (8, 0), and R (0, 0). A is the midpoint of B is the midpoint of ̶̶ QR. ̶̶ PR. Prove: AB =... |
�s name may come from its habit of pronking, or bouncing. When pronking, a springbok can leap up to 13 feet in the air. Springboks can run up to 53 miles per hour. 15. Recreation A hiking trail begins at E (0, 0). Bryan hikes from the start of the trail to a waterfall at W (3, 3) and then makes a 90° turn to a campsite... |
About It When you place two sides of a figure on the coordinate axes, what are you assuming about the figure? 26. This problem will prepare you for the Multi-Step TAKS Prep on page 280. Paul designed a doghouse to fit against the side of his house. His plan consisted of a right triangle on top of a rectangle. a. Find ... |
and (0, 0). What coordinates could be used so that a coordinate proof would be easier to complete? 34. Rectangle ABCD has dimensions of 2f and 2g units. The equation of the line containing ̶̶ g __ BD is y = x, and f ̶̶ g __ x + 2g. AC is y = - the equation of the line containing f Use algebra to show that the coordina... |
two sides of a triangle are congruent, then the angles opposite the sides are congruent. 4-8-2 Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. ∠B ≅ ∠C ̶̶ DE ≅ ̶̶ DF Theorem 4-8-1 is proven below. You will prove Theorem 4-8-2 in Exerc... |
September is 4.2 × 10 13 km, what is the distance from Earth to the star in March? Explain. E X A M P L E 2 Finding the Measure of an Angle Find each angle measure. A m∠C m∠C = m∠B = x° m∠C + m∠B + m∠A = 180 x + x + 38 = 180 2x = 142 x = 71 Thus m∠C = 71°. B m∠S Isosc. △ Thm. △ Sum Thm. Substitute the given values. Si... |
equiangular △ The measure of each ∠ of an equiangular △ is 60°. Subtract 15 from both sides. Divide both sides by 3. Equiangular △ → equilateral △ Def. of equilateral △ Subtract 2t and add 8 to both sides. t = 4.5 Divide both sides by 2. 3. Use the diagram to find JL. E X A M P L E 4 Using Coordinate Proof A coordinat... |
ABC are A (0, 2b), B (-2a, 0), and C (2a, 0). Prove △XYZ is isosceles. 4- 8 Isosceles and Equilateral Triangles 275 275 ������������������������������������������������������������ THINK AND DISCUSS 1. Explain why each of the angles in an equilateral triangle measures 60°. 2. GET ORGANIZED Copy and complete the graphic... |
to B, the angle to the plane is 80°. How can you find BT? � ������ ��� ��� � Find each angle measure. 13. m∠E 14. m∠TRU 15. m∠F ������������������ ��������� 16. m∠A Find each value. 17. z 18. y 19. BC 20. XZ 21. Given: △ABC is isosceles. P is the midpoint ̶̶ AB. Q is the midpoint of ̶̶ AC. of ̶̶ AB ≅ ̶̶ PC ≅ ̶̶ AC ̶̶ ... |
figure formed by (-2, 1), (5, 5), and (-1, -7). Estimate the measure of each angle and make a conjecture about the classification of the figure. Then use a protractor to measure each angle. Was your conjecture correct? Why or why not? 32. How many different isosceles triangles have a perimeter of 18 and sides whose le... |
right angles. ̶̶ ̶̶ DF, and AC ≅ Prove: △ABC ≅ △DEF ̶̶ AB ≅ ̶̶ DE. EF. Mark G so that FG = CB. Thus Proof: On △DEF draw ̶̶ DF and ∠C and ∠F are right angles. ̶̶ AC ≅ lines. Thus ∠DFG is a right angle, and ∠DFG ≅ ∠C. △ABC ≅ △DGF by SAS. ̶̶̶ ̶̶ DG ≅ DE by the Transitive Property. By the Isosceles Triangle Theorem ∠G... |
0) and B (a, b). What are all possible coordinates of the third vertex? SPIRAL REVIEW Find the solutions for each equation. (Previous course) 48. x 2 + 5x + 4 = 0 49. x 2 - 4x + 3 = 0 50. x 2 - 2x + 1 = 0 Find the slope of the line that passes through each pair of points. (Lesson 3-5) 51. (2, -1) and (0, 5) 52. (-5, -... |
enough wood for the 4 trusses of the doghouse? (Hint: You need to use the Pythagorean Theorem to find the two unknown side lengths of each truss.) 280 280 Chapter 4 Triangle Congruence ��������������� Quiz for Lessons 4-4 Through 4-8 4-4 Triangle Congruence: SSS and SAS SECTION 4B 1. The figure shows one tower and the... |
. M is the midpoint of ̶̶ JK, and N is the midpoint of ̶̶ KL. Prove: △KMN is isosceles. Ready to Go On? 281 281 �������������������������������������������� EXTENSION EXTENSION Proving Constructions Valid TEK G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures... |
above diagram Prove: CD is the perpendicular bisector of ̶̶ AB. 282 282 Chapter 4 Triangle Congruence ����� E X A M P L E 2 Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: ∠A ≅ ∠D To review the construction of an angle congruent to another angle, see page 22. Proof... |
............... 273 exterior.................... 225 remote interior angle....... 225 congruent polygons......... 231 exterior angle.............. 225 right triangle............... 216 coordinate proof............ 267 included angle.............. 242 scalene triangle............. 217 corollary................... 224 in... |
42 x = 14 m∠S = 6 (14) = 84° 284 284 Chapter 4 Triangle Congruence 7. In△LMN, m∠L = 8x °, m∠M = (2x + 1) °, and m∠N = (6x - 1) °. �������������������������������������������� 4-3 Congruent Triangles (pp. 231–237) TEKS G.2.B, G.10.B E X A M P L E EXERCISES ■ Given: △DEF ≅ △JKL. Identify all pairs of Given: △PQR ≅ △XYZ.... |
FH, ̶̶ GJ. ̶̶ RS ≅ ̶̶ VS ≅ ̶̶ UT ̶̶ VT 1. 2. 3. V is the mdpt. of ̶̶ UV ̶̶ RV ≅ 4. ̶̶ RU. 1. Given 2. Given 3. Given 4. Def. of mdpt. 14. Show that △ABC ≅ △XYZ when x = -6. 5. △RSV ≅ △UTV 5. SSS Steps 1, 2, 4 ■ Show that △ADB ≅ △CDB when s = 5. AB = s 2 - 4s AD = 14 - 2s 15. Show that △LMN ≅ △PQR when y = 25. = 5 2 - ... |
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