* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Proving Triangles Similar
Survey
Document related concepts
Transcript
7-3 Common Core State Standards Proving Triangles Similar G-SRT.B.5 Use . . . similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Also G-GPE.B.5 MP 1, MP 3, MP 4 Objectives To use the AA ∙ Postulate and the SAS ∙ and SSS ∙ Theorems To use similarity to find indirect measurements Are the triangles similar? How do you know? (Hint: Use a centimeter ruler to measure the sides of each triangle.) You’ve already learned how to decide whether two polygons are similar. This is a special case of that problem. MATHEMATICAL PRACTICES In the Solve It, you determined whether the two triangles are similar. That is, you needed information about all three pairs of angles and all three pairs of sides. In this lesson, you’ll learn an easier way to determine whether two triangles are similar. Lesson Vocabulary •indirect measurement Essential Understanding You can show that two triangles are similar when you know the relationships between only two or three pairs of corresponding parts. Postulate 7-1 Angle-Angle Similarity (AA =) Postulate Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. If . . . ∠S ≅ ∠M and ∠R ≅ ∠L L R S Then . . . △SRT ∙ △MLP T M P hsm11gmse_0703_t05714 450 Chapter 7 Similarity Problem 1 What do you need to show that the triangles are similar? To use the AA ∙ Postulate, you need to prove that two pairs of angles are congruent. Using the AA = Postulate V W Are the two triangles similar? How do you know? 45 A △RSW and △VSB S 45 ∠R ≅ ∠V because both angles measure 45°. ∠RSW ≅ ∠VSB because vertical angles are congruent. So, △RSW ∙ △VSB by the AA ∙ Postulate. B R B △JKL and △PQR ∠L ≅ ∠R because both angles measure 70°. By the Triangle Angle-Sum Theorem, m∠K = 180 - 30 - 70 = 80 and 30 J m∠P = 180 - 85 - 70 = 25. Only one pair of angles is congruent. So, △JKL and △PQR are not similar. K hsm11gmse_0703_t05715 Q 85 70 L R 70 P 1. Are the two triangles similar? How do you know? Got It? a. b. hsm11gmse_0703_t05716 39 62 68 51 Here are two other ways to determine whether two triangles are similar. hsm11gmse_0703_t05718 hsm11gmse_0703_t05717 Theorem 7-1 Side-Angle-Side Similarity (SAS =) Theorem Theorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar. If . . . Then . . . △ABC ∙ △QRS AB AC QR = QS and ∠A ≅ ∠Q A B Q C R S You will prove Theorem 7-1 in Exercise 35. Theorem 7-2 Side-Side-Side Similarity (SSS =) Theorem Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. hsm11gmse_0703_t05719 If ... Then . . . AB AC BC △ABC ∙ △QRS = = QR QS RS A B Q C R S You will prove Theorem 7-2 in Exercise 36. hsm11gmse_0703_t05720 Lesson 7-3 Proving Triangles Similar 451 Proof Proof of Theorem 7-1: Side-Angle-Side Similarity Theorem A AB AC Given: QR = QS , ∠A ≅ ∠Q Prove: △ABC ∙ △QRS B C R Plan for Proof: Choose X on RQ so that QX = AB. Draw < > XY } RS. Show that △QXY ∙ △QRS by the AA ∙ Postulate. Then use the proportion AB QR = AC QS to show that QX QR = QY QS S A Q and the given proportion AC = QY. Then prove that △ABC ≅ △QXY. hsm11gmse_0703_t05721 Y B C X Finally, prove that △ABC ∙ △QRS by the AA ∙ Postulate. Problem 2 Q R S hsm11gmse_0703_t05722 Verifying Triangle Similarity Are the triangles similar? If so, write a similarity statement for the triangles. A S 10 U V 6 9 T 8 12 W 15 X S Use the side lengths to identify corresponding sides. Then set up ratios for each pair of corresponding sides. 10 ST 6 2 hsm11gmse_0703_t05724 Shortest sides XV = 9 = 3 US 10 2 Longest sides WX = 15 = 3 TU 8 2 Remaining sides VW = 12 = 3 U 8 6 12 V 9 W 15 T X All three ratios are equal, so corresponding sides are proportional. hsm11gmse_0703_t05723 △STU ∙ △XVW by the SSS ∙ Theorem. How can you make it easier to identify corresponding sides and angles? Sketch and label two separate triangles. B L 2 M 8 K 12 P 3 N ∠K ≅ ∠K by the Reflexive Property of Congruence. 8 KL KP 12 4 4 hsm11gmse_0703_t05725 KM = 10 = 5 and KN = 15 = 5 . So, △KLP ∙ △KMN by the SAS ∙ Theorem. 452 Chapter 7 Similarity M L 8 2 10 8 K 12 P K 12 3 15 ∠K is the included angle between two known sides in each triangle. hsm11gmse_0703_t05726 N 2.Are the triangles similar? If so, write a similarity statement for the triangles Got It? and explain how you know the triangles are similar. 9 a. A b. A 8 L 8 G B C 6 C 8 6 E 6 8 12 W F 6 E Proof Proving Triangles Similar Problem 3hsm11gmse_0703_t05727 hsm11gmse_0703_t05728 J Statements You need to show that the triangles are similar. L H F Prove: △FGH ∙ △JKL The triangles are isosceles, so the base angles are congruent. K G Given: FG ≅ GH, JK ≅ KL, ∠F ≅ ∠J Find two pairs of corresponding hsm11gmse_0703_t05729 congruent angles and use the AA ∙ Postulate to prove the triangles are similar. Reasons 1) FG ≅ GH, JK ≅ KL 1) Given 2) △FGH is isosceles. 2) Def. of an isosceles △ △JKL is isosceles. 3) ∠F ≅ ∠H, ∠J ≅ ∠L 3) Base ⦞ of an isosceles △ are ≅. 4) ∠F ≅ ∠J 4) Given 5) ∠H ≅ ∠J 5) Transitive Property of ≅ 6) ∠H ≅ ∠L 6) Transitive Property of ≅ 7) △FGH ∙ △JKL 7) AA ∙ Postulate 3.a.Given: MP } AC Got It? Prove: △ABC ∙ △PBM b.Reasoning For the figure at the right, suppose you are given only that CA CB PM = MB . Could you prove that the triangles are similar? Explain. P C A B M hsm11gmse_0703_t05730 Lesson 7-3 Proving Triangles Similar 453 Essential Understanding Sometimes you can use similar triangles to find lengths that cannot be measured easily using a ruler or other measuring device. You can use indirect measurement to find lengths that are difficult to measure directly. One method of indirect measurement uses the fact that light reflects off a mirror at the same angle at which it hits the mirror. Problem 4 Finding Lengths in Similar Triangles Rock Climbing Before rock climbing, Darius wants to know how high he will climb. He places a mirror on the ground and walks backward until he can see the top of the cliff in the mirror. What is the height of the cliff? J x ft H 5.5 ft T Before solving for x, verify that the triangles are similar. △HTV ∙ △ JSV by the AA∙ Postulate because ∠T ≅ ∠S and ∠HVT ≅ ∠JVS. 6 ft V 34 ft S △HTV ∙ △JSV AA ∙ Postulate HT TV Corresponding sides of ∙ triangles are proportional. JS = SV 5.5 6 x = 34 Substitute. 187 = 6x Cross Products Property 31.2 ≈ x Solve for x. The cliff is about 31 ft high. 4.Reasoning Why is it important that the ground be flat to use the method of Got It? indirect measurement illustrated in Problem 4? Explain. 454 Chapter 7 Similarity Lesson Check MATHEMATICAL Do you know HOW? Do you UNDERSTAND? Are the triangles similar? If yes, write a similarity statement and explain how you know they are similar. 4.Vocabulary How could you use indirect measurement to find the height of the flagpole at your school? 1. R A 100 35 E 5.Error Analysis Which solution for the value of x in the figure at the right is not correct? Explain. 45 Z B 2. A 2 6 16 70 G 4 8 8 x 6 = 46 48 = 4x hsm11gmse_0703_t05734 12 = x 15 F 20 6.a.Compare and Contrast How are the SAS Similarity Theorem and the SAS Congruence hsm11gmse_0703_t05736 Postulate alike? How are they different? hsm11gmse_0703_t05735 b.How are the SSS Similarity Theorem and the SSS Congruence Postulate alike? How are they different? E hsm11gmse_0703_t05733 MATHEMATICAL Practice and Problem-Solving Exercises A Practice B. = 8x 4x = 72 x = 18 B 12 hsm11gmse_0703_t05732 70 A A. x 8 6 3 9 4 F 4 C hsm11gmse_0703_t05731 B 3 D E 4.5 3. U PRACTICES PRACTICES Determine whether the triangles are similar. If so, write a similarity statement and name the postulate or theorem you used. If not, explain. 7. F 8. J D M 8 See Problems 1 and 2. 9. R A 12 P 24 S 6 6 H G K 10. P 30 J 40 P R 11. R 10 U N N 22 T R 12. A 16 Q 8 24 B 35 60 G hsm11gmse_0703_t05737.ai 18 32 hsm11gmse_0703_t05738.ai hsm11gmse_0703_t05739.ai 22 Q K 45 L hsm11gmse_0703_t05740.ai S 25 Y 11 9 35 110 A C H 12 K Lesson 7-3 Proving Triangles Similar 455 hsm11gmse_0703_t05742.ai hsm11gmse_0703_t05741.ai 13.Given: ∠ABC ≅ ∠ACD 14.Given: PR = 2NP, See Problem 3. Proof Prove: △ABC ∙ △ACD Proof PQ = 2MP Prove: △MNP ∙ △QRP C R M A D N B P Q See Problem 4. Indirect Measurement Explain why the triangles are similar. Then find the distance represented by x. hsm11gmse_0703_t05743.ai 15. hsm11gmse_0703_t05744.ai 16. x x 90 ft 5 ft 6 in. 135 ft 120 ft 4 ft 10 ft Mirror Washington Monument At a certain time of day, a 1.8-m-tall person standing 17. next to the Washington Monument casts a 0.7-m shadow. At the same time, the Washington Monument casts a 65.8-m shadow. How tall is the Washington Monument? B Apply Can you conclude that the triangles are similar? If so, state the postulate or theorem you used and write a similarity statement. If not, explain. 18. A 18 E F 48 32 38 19. L 24 S D B 24 C 20. P N T M D 16 N 9 12 K 12 G 21.a.Are two isosceles triangles always similar? Explain. b.Are two right isosceles triangles always similar? Explain. hsm11gmse_0703_t05746.ai 22. Think About a Plan On a sunny day, a classmate uses indirect measurement to find hsm11gmse_0703_t05745.ai hsm11gmse_0703_t05747.ai the height of a building. The building’s shadow is 12 ft long and your classmate’s shadow is 4 ft long. If your classmate is 5 ft tall, what is the height of the building? • Can you draw and label a diagram to represent the situation? • What proportion can you use to solve the problem? 23. Indirect Measurement A 2-ft vertical post casts a 16-in. shadow at the same time a nearby cell phone tower casts a 120-ft shadow. How tall is the cell phone tower? 456 Chapter 7 Similarity Algebra For each pair of similar triangles, find the value of x. 24. 25. 26. 2x 4 2x 10 39 x 8 9 5 x2 x x6 x 14 24 PQ QR 28.Given: AB } CD, BC } DG Prove: AB CG = CD AC 27.Given: PQ # QT, ST # TQ, ST = TV Proof Prove: △VKR is isosceles. hsm11gmse_0703_t05748.ai P S # Proof # hsm11gmse_0703_t05750.ai hsm11gmse_0703_t05749.ai B D K Q V R A T G C 29. Reasoning Does any line that intersects two sides of a triangle and is parallel to the third side of the triangle form two similar triangles? Justify your reasoning. hsm11gmse_0703_t05752.ai 30. Constructions Draw any △ABC with m∠C = 30. Use ahsm11gmse_0703_t05751.ai straightedge and compass to construct △LKJ so that △LKJ ∙ △ABC. R W P 31. Reasoning In the diagram at the right, △PMN ∙ △SRW . MQ T Q and RT are altitudes. The scale factor of △PMN to △SRW is 4 : 3. What is the ratio of MQ to RT ? Explain how you know. N M 32.Coordinate Geometry △ABC has vertices A(0, 0), B(2, 4), and Proof C(4, 2). △RST has vertices R(0, 3), S( -1, 5), and T( -2, 4). Prove that △ABC ∙ △RST . (Hint: Graph △ABC and △RST in the coordinate plane.) 33.Write a proof of the following: Any two nonvertical parallel lines have equal slopes. y 1 hsm11gmse_0703_t05753.ai E Proof 2 B Given: Nonvertical lines /1 and /2, /1 } /2 , EF and BC are # to the x-axis BC EF Prove: AC = DF S x O A C D F 34.Use the diagram in Exercise 33. Prove: Any two nonvertical lines with equal slopes are parallel. Proof C Challenge 35.Prove the Side-Angle-Side Similarity Theorem (Theorem 7-1). Proof AB AC Given: QR = QS , ∠A ≅ ∠Q Prove: △ABC ∙ △QRS A hsm11gmse_0703_t05754.ai Q B C R S hsm11gmse_0703_t05755.ai Lesson 7-3 Proving Triangles Similar 457 36.Prove the Side-Side-Side Similarity Theorem (Theorem 7-2). A Proof Q AB AC BC Given: QR = QS = RS Prove: △ABC ∙ △QRS B C R S Standardized Test Prep SAT/ACT 37.Complete the statement △ABC ∙ ? . By which postulate or theorem are the triangles similar? △AKN ; SSS ∙ △ANK ; SAS ∙ △AKN ; SAS ∙ △ANK ; AA ∙ hsm11gmse_0703_t05756.ai N B 2 8 A C 3 K 12 38.∠1 and ∠2 are alternate interior angles formed by two parallel lines and a transversal. If m∠2 = 68, what is m∠1? 122 hsm11gmse_0703_t05757.ai 39.The length of a rectangle is twice its width. If the perimeter of the rectangle is 72 in., what is the length of the rectangle? Extended Response 22 68 112 12 in. 18 in. 24 in. 36 in. 40.Graph A(2, 4), B(4, 6), C(6, 4), and D(4, 2). What type of polygon is ABCD? Justify your answer. Mixed Review TRAP = EZYD. Use the diagram at the right to find the following. 41. the scale factor of TRAP to EZYD 44. DE PT 42. m∠R 43.DY T 4 R Y D 45 4 P 8 A Z 6 E See Lesson 1-4. Use a protractor to find the measure of each angle. Classify the angle as acute, right, obtuse, or straight. hsm11gmse_0703_t05758.ai 45. 46. See Lesson 7-2. 47. 48. Get Ready! To prepare for Lesson 7- 4, do Exercises 49 –52. hsm11gmse_0703_t05761.ai See Lesson 7-1. Algebra Identify the means and extremes of each proportion. Then solve for x. hsm11gmse_0703_t05759.ai hsm11gmse_0703_t05760.ai x 18 15 9 x-3 5 18 49. 50. 12 51. x + 2 = x 52.xhsm11gmse_0703_t05762.ai +4=9 8 = 24 m = 20 458 Chapter 7 Similarity