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4-5 4-5 Isosceles and Equilateral Triangles 1. Plan Objectives 1 To use and apply properties of isosceles triangles Examples 1 2 3 Using the Isosceles Triangle Theorems Using Algebra Real-World Connection What You’ll Learn • To use and apply properties of isosceles triangles . . . And Why Lesson 3-4 B x⬚ 3. Name the side opposite &A. BC To find the angles of a garden path, as in Example 3 4. Name the side opposite &C. BA A x 2 5. Algebra Find the value of x. 105 75⬚ 30⬚ C New Vocabulary • legs of an isosceles triangle • base of an isosceles triangle • vertex angle of an isosceles triangle • base angles of an isosceles triangle • corollary Math Background Understanding the vocabulary, including legs, vertex angle, and base angles, is necessary for solving many exercises in this section. Help students remember the meanings of terms by having them describe everyday objects with the same words. Mention that isosceles derives from the Greek iso (same) and skelos (leg). Trapezoids, which will be studied in Chapter 4, can also be isosceles, and have two bases. GO for Help Check Skills You’ll Need 1. Name the angle opposite AB. lC 2. Name the angle opposite BC. lA 1 The Isosceles Triangle Theorems Vocabulary Tip Isosceles is derived from the Greek isos for equal and skelos for leg. Vertex Angle Isosceles triangles are common in the real world. You can find them in structures such as bridges and buildings. The congruent sides of an isosceles triangle are its legs. The third side is the base. The two congruent sides form the vertex angle. The other two angles are the base angles. Legs Base Base Angles An isosceles triangle has a certain type of symmetry about a line through its vertex angle. The theorems below suggest this symmetry, which you will study in greater detail in Lesson 9-4. More Math Background: p. 196D Key Concepts Lesson Planning and Resources Theorem 4-3 Isosceles Triangle Theorem C If two sides of a triangle are congruent, then the angles opposite those sides are congruent. See p. 196E for a list of the resources that support this lesson. &A > &B Theorem 4-4 A Converse of Isosceles Triangle Theorem B C PowerPoint If two angles of a triangle are congruent, then the sides opposite the angles are congruent. Bell Ringer Practice AC > BC Check Skills You’ll Need A B For intervention, direct students to: Theorem 4-5 Using Exterior Angles of Triangles CD ' AB and CD bisects AB. 228 A D B Chapter 4 Congruent Triangles Special Needs Below Level L1 Have students collect other landscape designs from gardening and landscaping books and magazines. Have them select one design and describe it using geometric terms. 228 C The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Lesson 3-4: Example 3 Extra Skills, Word Problems, Proof Practice, Ch. 3 learning style: tactile L2 Discuss whether the equilateral-equiangular relationship holds for polygons with more than 3 sides. Ask students to support their reasoning with examples. learning style: verbal Proof In the following proof of the Isosceles Triangle Theorem, you use a special segment, the bisector of the vertex angle. You will prove Theorems 4-4 and 4-5 in the Exercises. 2. Teach Proof of the Isosceles Triangle Theorem Guided Instruction Begin with isosceles #XYZ with XY > XZ. Draw XB, the bisector of the vertex angle &YXZ. X Connection to Chemistry 1 2 Given: XY > XZ, XB bisects &YXZ. Ask: What are different forms of a chemical element with the same atomic number called? isotopes Point out that the prefix iso-, meaning “equal,” is a variation of the prefix isos- in isosceles. Prove: &Y > &Z Real-World B Y Proof: You are given that XY > XZ. By the definition of angle bisector, &1 > &2. By the Reflexive Property of Congruence, XB > XB. Therefore, by the SAS Postulate, #XYB > #XZB, and &Y > &Z by CPCTC. Z Visual Learners Students may think that the base of an isosceles triangle is always at the bottom. Illustrate by rotating a physical model that the base can be in any orientation, as in Exercise 30. Connection 1 This A-shaped roof has congruent legs and congruent base angles. EXAMPLE Using the Isosceles Triangle Theorems T Explain why each statement is true. a. &WVS > &S U WV > WS so &WVS > &S by the Isosceles Triangle Theorem. b. TR > TS V &R > &WVS and &WVS > &S, so &R > &S by the Transitive Property of >. TR > TS by the Converse of the Isosceles Triangle Theorem. Quick Check 2 A 1 2 A 4 5 A B B D C C Using Algebra M Multiple Choice Find the value of y. 17 27 54 90 E D C B E D C B A 3 EXAMPLE E D C B A D E E Test-Taking Tip Remember that the acute angles of a right triangle are complementary. S 1 Can you deduce that #RUV is isosceles? Explain. No, point U could be anywhere between R and T. E D C B Alternative Method W R By Theorem 4-5, you know that MO ' LN, so &MON = 90. #MLN is isosceles, so &L > &N and m&N = 63. m&N + 90 + y = 180 Triangle Angle-Sum Theorem 63 + 90 + y = 180 Substitute for mlN and x. y = 27 y⬚ N 63⬚ O L 1 3 2 Find the value of x. 6 A corollary is a statement that follows immediately from a theorem. Corollaries to the Isosceles Triangle Theorem and its converse appear on the next page. Lesson 4-5 Isosceles and Equilateral Triangles Advanced Learners 229 English Language Learners ELL L4 Have students explain why Theorems 4-3, 4-4, and 4-5 do or do not apply to equilateral triangles. learning style: verbal EXAMPLE Math Tip Point out that every corollary is a theorem that can be proved. Subtract 153 from each side. The correct answer is B. Quick Check Draw identical copies of XYZ side by side. Then label congruent parts, asking the class to justify each step. • Use a single tick mark to show XY on the first copy XZ on the second copy. • Use an arc to show &X on the first copy &X on the second copy. • Use double tick marks to show XZ on the first copy XY on the second copy. Ask: Which triangles are congruent? By what postulate? kXYZ O kXZY; SAS Which angles are congruent by CPCTC? lY and lZ EXAMPLE Teaching Tip Have students calculate the total measure of the four angles formed by the x-axis and the y-axis at the origin. Discuss the fact that there are 360° about any point. Ask students to suggest other ways to show that there are 360° about any point, such as drawing a straight angle and adding the measures on each side. Point out that equilateral and equiangular can be used interchangeably for triangles, but not for other polygons. For example, squares and rectangles are both equiangular but only the square is equilateral. learning style: verbal 229 PowerPoint Key Concepts Additional Examples B Vocabulary Tip &X > &Y > &Z Equilateral: Congruent sides Equiangular: Congruent angles A Corollary to Theorem 4-3 Y If a triangle is equilateral, then the triangle is equiangular. 1 Explain why ABC is isosceles. X Corollary Corollary X Corollary to Theorem 4-4 Z Y If a triangle is equiangular, then the triangle is equilateral. C XY > YZ > ZX X Z * ) * ) Because XA n BC , lABC O lXAB. By the angles marked O and the Transitive Prop., lABC O lACB. kABC is isosceles by Converse of the Isosceles Triangle Thm. 2 Use the diagram for Example 2. Suppose that m&L 2 63 and m&L = y. Find the values of x and y. x ≠ 90, y ≠ 45 3 x˚ In a rectangle, an angle measure is 90; in an equilateral triangle, it is 60. For: Isosceles Triangle Activity Use: Interactive Textbook, 4-5 Quick Check x + 90 + 60 + 90 = 360 x = 120 3 What is the measure of the angle at each outside corner of the path? 150 EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A Resources • Daily Notetaking Guide 4-5 L3 • Daily Notetaking Guide 4-5— L1 Adapted Instruction Connection Landscaping A landscaper uses rectangles and equilateral triangles for the path around the hexagonal garden. Find the value of x. 3 In the drawing for Example 3, suppose that a segment is drawn between the endpoints of the segments that determine the angle marked x°. Find the angle measures of the triangle that is formed. 120, 30, 30 Real-World EXAMPLE Practice by Example Example 1 GO for Help (page 229) Complete each statement. Explain why it is true. V 1. VT > 9 2. UT > 9 > YX 3. VU > 9 U 4. &VYU > 9 T Closure W Example 2 x 2 Algebra Find the values of x and y. An angle exterior to the vertex of an isosceles triangle measures 80. Find the angle measures of the triangle. 100, 40, 40 (page 229) 5. 6. 7. x⬚ x⬚ Example 3 (page 230) x⬚ 100⬚ 50⬚ y⬚ 1. VX ; Converse of the Isosc. kThm. 2. UW ; Converse of the Isosc. kThm. 3. VY ; VT ≠ VX (Ex. 1) and UT ≠ YX (Ex. 2), so Y VU ≠ VY by the Subtr. Prop. of ≠. X 4. Answers may vary. Sample: lVUY; ' opposite O sides are O. 110⬚ y⬚ 4 52⬚ y x ≠ 38; y ≠ 4 x ≠ 40; y ≠ 70 x ≠ 80; y ≠ 40 8. A square and a regular hexagon are placed so that they have a common side. Find m&SHA and m&HAS. 150; 15 H 9. Five fences meet at a point to form angles with measures x, 2x, S 3x, 4x, and 5x around the point. Find the measure of each angle. 24, 48, 72, 96, 120 230 230 A Chapter 4 Congruent Triangles 18. Answers may vary. Sample: Corollary to Thm. 4-3: Since XY O YZ , lX O lZ by Thm. 4-3. YZ O ZX , so lY O lX by Thm. 4-3 also. By the Trans. Prop., lY O lZ, so lX O lY O lZ. Corollary to Thm. 4-4: Since lX O lZ, XY O YZ by Thm. 4-4. lY O lX, so YZ O ZX by Thm. 4.4 also. By the Trans Prop., XY O ZX , so XY O YZ O ZX . 26. No; the k can be positioned in ways such that the base is not on the bottom. B Apply Your Skills Find each value. 3. Practice K 10. If m&L = 58, then m&LKJ = 7. 64 Assignment Guide 11. If JL = 5, then ML = 7. 2 12 12. If m&JKM = 48, then m&J = 7. 42 13. If m&J = 55, then m&JKM = 7. 35 1 A B 1-32 J M L 14. Architecture Seventeen spires, pictured at the left, cover the Cadet Chapel at the Air Force Academy in Colorado Springs, Colorado. Each spire is an isosceles triangle with a 40° vertex angle. Find the measure of each base angle. 70 15. Developing Proof Here is another way to prove the Isosceles Triangle Theorem. Supply the missing parts. K Begin with isosceles #HKJ with KH > KJ. Draw a. 9, a bisector of the base HJ. KM H Statements 1. 2. 3. 4. 5. 6. Exercise 14 Proof KM bisects HJ. HM > JM KH > KJ KM > KM #KHM > #KJM &H > &J M Test Prep Mixed Review 41-44 45-48 Homework Quick Check To check students’ understanding of key skills and concepts, go over Exercises 6, 8, 17, 18, 28. Exercise 14 The world’s tallest J cathedral spire, measuring 528 ft, is on the Protestant Cathedral of Ulm in Germany. Construction began in 1377, but the spire was not finished until 1890. Reasons c. d. 3. e. f. g. 33-40 Connection to Architecture Given: KH > KJ, b. 9 bisects HJ. KM Prove: &H > &J C Challenge 9 By construction 9 Def. of segment bisector Given 9 Reflexive Prop. of O 9 SSS 9 CPCTC 16. Supply the missing parts in this statement of the Converse of the Isosceles Triangle Theorem. Then write a proof. Exercise 19 To avoid careless errors, encourage students to show the solution steps for solving their algebraic equation. R Begin with #PRQ with &P > &Q. Draw a. 9, the bisector of &PRQ. RS Given: &P > &Q, b. 9 bisects &PRQ. RS Prove: PR > QR See back of book. 17a. 30, 30, 120 S P Q 17. Graphic Arts A former logo for GPS the National Council of Teachers of Mathematics is shown at the right. Trace the logo onto paper. a. Highlight an obtuse isosceles The triangles in the logo have these congruent triangle in the design. Then sides and angles. find its angle measures. See left. b. How many different sizes of angles can you find in the logo? What are their measures? 5; 30, 60, 90, 120, 150 GPS Guided Problem Solving nline Homework Help Visit: PHSchool.com Web Code: aue-0405 L2 Reteaching L1 Adapted Practice Practice Name Class L3 Date Practice 4-5 Isosceles and Equilateral Triangles Find the values of the variables. 1. 2. 3. x 110 t 10 y x y 5. s X 6. WXYZV is a regular polygon. y b W Y r 125 x x 19. Multiple Choice The perimeter of the triangle at the right is 20. Find x. C 3 5 6 7 Lesson 4-5 Isosceles and Equilateral Triangles z 7. 8. c Z V a 9. c a 30 x z 60 b 2x 6 2x ⫺ 5 Complete each statement. Explain why it is true. 10. AF 9 11. CA 9 231 © Pearson Education, Inc. All rights reserved. GO L4 Enrichment 4. 18. Writing Explain how each corollary on page 230 follows from its theorem. First, write one explanation and then write the second similar to the first. See margin. L3 12. KI 9 A K 13. EC 9 B J 14. JA 9 I G 15. HB 9 C H F L D E Given mlD ≠ 25, find the measure of each angle. 16. JAB 17. FAL 18. JKI 19. DLA Find the values of x and y. 20. y x 55 21. x 110 22. y y x 231 4. Assess & Reteach x 2 Algebra Find the values of x and y. 20. D 21. PowerPoint 60⬚ Lesson Quiz B 1. If m&BAC = 38, find m&C. 71 2. If m&BAM = m&CAM = 23, find m&BMA. 90 3. If m&B = 3x and m&BAC = 2x - 20, find x. 25 4. Real-World 18 60° Connection Careers Radio broadcasters must respond to opinions given by “call-in” listeners. 0 27. Reasoning What are the measures 450 ft 550 ft of the base angles of an isosceles Tower cables extend to both widths. right triangle? Explain. 45; they are ≠ and have sum 90. y 60° 60° x° Find the values of x and y. x ≠ 60; y ≠ 9 28. lA O lD by the Isos. k Thm. kABE O kDCE by SAS. Proof 28. Given: AE > DE, AB > DC Proof 29. Prove Theorem 4-5. Use the diagram next to it on page 228. See margin. 30. 126⬚ C m⬚ E A B C D x 2 Algebra Find the values of m and n. B F E Prove: #ABE > #DCE 5. ABCDEF is a regular hexagon. Find m&BAC. A 30⬚ 24. Critical Thinking An exterior angle of an isosceles triangle has measure 100. Find two possible sets of measures for the angles of the triangle. 80, 80, 20; 80, 50, 50 25. a. Communications In the diagram isosc. > at the right, what type of triangles Radio 1000 are formed by the cables of the Tower ft 1009 ft tall same height and the ground? b. What are the two different base 800 lengths of the triangles? 900 ft; 1100 ft Cables c. How is the tower related to 600 each of the triangles? The tower is the ' bis. of the base of each k. 400 26. Critical Thinking Curtis defines the base of an isosceles triangle as its “bottom side.” Is his definition a 200 good one? Explain. See margin. C M y⬚ x⬚ A B x ≠ 60; y ≠ 30 x ≠ 30; y ≠ 120 ABCDE is a regular pentagon. x ≠ 36; y ≠ 36 23. Write the Isosceles Triangle Theorem and its converse as a biconditional. 23. Two sides of a k are O if and only if the ' opp. those sides are O. A C y⬚ y⬚ Use the diagram for Exercises 1–3. x⬚ E x⬚ 22. D C Alternative Assessment 33. (4, 0) and (0, 4) 35. (5, 3); (2, 6); (2, 9); (8, 3); (–1, 6); (5, 0)) 29. AC O CB and lACD O lDCB are given. CD O CD by the Refl. Prop. of 232 O, so kACD O kBCD by 232 50⬚ m⬚ m⬚ n⬚ 34. (0, 0) and (5, 5) 35. (2, 3) and (5, 6) x 2 36. Algebra A triangle has angle measures x + 15, 3x - 35, and 4x. 34. (5, 0); (0, 5); (–5, 5); (5, –5); (0, 10); (10, 0) The diagram illustrates the Isosceles Triangle Theorem. Have the class draw similar diagrams to illustrate the Converse of the Isosceles Triangle Theorem and the two corollaries from this lesson. n⬚ m ≠ 60; n ≠ 30 m ≠ 20; n ≠ 45 Coordinate Geometry For each pair of points, there are six points that could be the third vertex of an isosceles right triangle. Find the coordinates of each point. Challenge 33. (0, 0), (4, 4), (–4, 0), (0, –4), (8, 4), (4, 8) 32. n⬚ m ≠ 36; n ≠ 27 30 Draw the diagram below on the board. 31. a. Find the value of x. 25 b. Find the measure of each angle. 40; 40; 100 c. What type of triangle is it? Why? Obtuse isosc. > : 2 of the ' are O and one l is obtuse. 37. State the converse of Theorem 4-5. If the converse is true, write a paragraph proof. If the converse is false, give a counterexample. See margin. Chapter 4 Congruent Triangles SAS. So AD O DB by CPCTC, and CD bisects AB. Also lADC O lBDC by CPCTC, mlADC ± mlBDC ≠ 180 by l Add. Post., so mlADC ≠ mlBDC ≠ 90 by the Subst. Prop. So CD is the # bis. of AB. 37. The # bis. of the base of an isosc. k is the bis. of the vertex l; given isosc. kABC with # bis. CD, lADC O lCDB and AD O DB by def. of # bis. Since CD O CD by Refl. Prop., kACD O kBCD by SAS. So lACD O lBCD by CPCTC, and CD bisects lACB. Real-World Connection The circle is a basic shape for many tile designs. 38. Crafts The design in Step 3 is used in Hmong crafts and in Islamic and Mexican tiles. To create it, the artist starts by drawing a circle and four equally spaced diameters. a. How many different sizes of isosceles right Step 1 Step 2 Step 3 triangles can you find in Step 2? Trace an example of each onto your paper. 5 b. How many times does a triangle of each size in part (a) appear in the Step 2 diagram? See back of book. Test Prep Resources For additional practice with a variety of test item formats: • Standardized Test Prep, p. 253 • Test-Taking Strategies, p. 248 • Test-Taking Strategies with Transparencies Reasoning What measures are possible for the base angles of each type of triangle? Explain. 39. an isosceles obtuse triangle 0 R measure of base l R 45 40. an isosceles acute triangle 45 R measure of base l R 90 Test Prep Multiple Choice 41. In isosceles #ABC, the vertex angle is &A. What can be proved? C A. AB = CB B. &A > &B C. m&B = m&C D. BC > AC 1 42. In the diagram at the right, m&1 = 40. What is m&2? G F. 40 G. 50 H. 80 J. 100 2 43. In an isosceles triangle, the measure of the vertex angle is 4x. The measure of each base angle is 2x + 10. What is the measure of the vertex angle? D A. 10 B. 20 C. 50 D. 80 Short Response 44. In the figure at the right, m&APB = 60. a. What is m&PAB? Explain. b. &PAB and &QAB are complementary. What is m&AQB? Show your work. a–b. See margin. A P Q B Mixed Review 45. m&R = 59, m&T = 93 = m&H, m&V = 28, and RT = GH. What, if anything, can you for conclude about RC and GV? Explain. Help RC ≠ GV; RC O GV by CPCTC since kRTC O kGHV by ASA. R Lesson 4-4 GO Lessons 4-2, 4-3 V G C H Which congruence statement, SSS, SAS, ASA, or AAS, would you use to conclude that the two triangles are congruent? 46. Lesson 3-5 T AAS 47. SSS 48. How many sides are in a regular polygon whose exterior angles measure 158? 24 sides lesson quiz, PHSchool.com, Web Code: aua-0405 [2] a. 60; since mlPAB ≠ mlPBA, and mlPAB ± mlPBA ≠ 120, mlPAB ≠ 60. Lesson 4-5 Isosceles and Equilateral Triangles 233 b. 120; mlAPB ≠ 60 so mlPAB ≠ 60. Since lPAB and lQAB are compl., mlQAB ≠ 30. kQAB is isosc. so mlAQB ≠ 120. [1] one part correct 233