# 4-6 Isosceles and Equilateral Triangles. Refer to the figure. 1. If name two congruent angles. ANSWER: BAC and BCA

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1 Refer to the figure. 1. If name two congruent angles. BAC and BCA 2. If EAC ECA, name two congruent segments PROOF Write a two-column proof. Given: is isosceles; bisects ABC. Prove: Find each measure. 3. FH m MRP 1. is isosceles; bisects ABC. (Given) 2. (Def. of isosceles) 3. ABE CBE (Def. of bisector) 4. (Refl. Prop.) 5. (SAS) 60 ANALYZE RELATIONSHIPS Find the value of each variable esolutions Manual - Powered by Cognero Page 1

2 8. ROLLER COASTERS A roller coaster track appears to be composed of congruent triangles. A portion of the track is shown. a. If and are perpendicular to is Refer to the figure. isosceles with base, and prove that b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning. 9. If name two congruent angles. ABE and AEB 10. If ABF AFB, name two congruent segments. 11. If name two congruent angles. a. Given: and are perpendicular to Prove: is isosceles with base and 1. and are perpendicular to is isosceles with base and (Given) 2. RQV and STV are right angles. (Def. of ) 3. RQV STV (All rt. s are.) 4. (Def. of isosceles) 5. VSR VRS (Isos. Thm.) 6. QVR VRS TVS VSR (Alt. Int. s Thm.) 7. TVS QVR (Trans. Property) 8. (AAS) b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have = QT. So QT = 3 m. ACD and ADC 12. If DAE DEA, name two congruent segments. 13. If BCF BFC, name two congruent segments. 14. If name two congruent angles. AFH and AHF Find each measure. 15. m BAC 60 esolutions Manual - Powered by Cognero Page 2

3 16. m SRT x = 11, y = TR CB 22. x = 1.4, y = 1 PROOF Write a paragraph proof. 23. Given: is isosceles and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL ANALYZE RELATIONSHIPS Find the value of each variable. x = 5 We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Since is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL. 20. x = 13 esolutions Manual - Powered by Cognero Page 3

5 26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC. Refer to the figure on page CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. Given: and is an isosceles with base Prove: bisects the angle formed by the sloped sides of the roof, ABC. 1. and is an isosceles with base (Given) 2. BDA and BDC are rt. s. (Def. of ) 3. BDA BDC (All rt. s are ) 4. (Def. of Isos.) 5. BAD BCD (Isos. Thm.) 6. (AAS) 7. ABD CBD (CPCTC) 8. bisects the angle formed by the sloped sides of the roof, ABC. (Def. of bisector) Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Since AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles. 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. Conjecture: The measures of the base angles of an isosceles right triangle are 45. The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. esolutions Manual - Powered by Cognero Page 5

6 ANALYZE RELATIONSHIPS Find each measure. 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F. 29. m CAD m ACD m ACB m ABC 22 Given: Each triangle is isosceles, G H, and H J. Prove: The distance from B to F is three times the distance from D to F. 1. Each triangle is isosceles, G H, and H J. (Given) 2. G J (Trans. Prop.) 3. (Def. of Isosceles) 4. (Trans. Prop.) 5. (Trans. Prop.) 6. (Trans. Prop.) 7. (SAS) 8. (CPCTC) 9. BC = CD = DF (Def. of congruence) 10. BC + CD + DF = BF (Seg. Add. Post.) 11. DF + DF + DF = BF (Subst.) 12. 3DF = BF (Addition) esolutions Manual - Powered by Cognero Page 6

7 34. Given: is isosceles; Prove: X and YZV are complementary. PROOF Write a two-column proof of each corollary or theorem. 35. Corollary is isosceles; (Given) 2. X WVX (Isos. Thm.) 3. WVX YVZ (Vert s are.) 4. X YVZ (Trans. Prop.) 5. m X = m YVZ (Def. of s) 6. m VYZ = 90 ( lines form rt s.) 7. is a right triangle. (Def of rt. ) 8. YZV and YVZ are complementary. (The acute s of a rt. are comp.) 9. m YZV + m YVZ = 90 (Def. Of Compl. s) 10. m YZV + m X = 90 (Subst.) 11. X and YZV are complementary (Def. of Compl. s) Case I Given: is an equilateral triangle. Prove: is an equiangular triangle. 1. is an equilateral triangle. (Given) 2. (Def. of equilateral 3. A B C (Isosceles Th.) 4. is an equiangular triangle. (Def. of equiangular) Case II Given: is an equiangular triangle. Prove: is an equilateral triangle. 1. is an equiangular triangle. (Given) 2. A B C (Def. of equiangular ) 3. (If 2 s of a are then the sides opp. those s are ) 4. is an equilateral triangle. (Def. of equilateral) esolutions Manual - Powered by Cognero Page 7

8 36. Corollary 4.4 Given: is an equilateral triangle. Prove: m A = m B = m C = Find the value of each variable is an equilateral triangle. (Given) 2. (Def. of equilateral ) 3. A B C (Isosceles Thm.) 4. m A = m B = m C (Def. of s) 5. m A + m B + m C = 180 (Triangle Angle Sum Thm.) 6. 3m A = 180 (Subst.) 7. m A = 60 (Div. Prop.) 8. m A = m B = m C = 60 (Subst.) 37. Theorem 4.11 Given: A C Prove: GAMES Use the diagram of a game timer shown to find each measure. 40. m LPM Let bisect ABC. (Protractor Post.) 2. ABD CBD (Def. of bisector) 3. A C (Given) 4. (Refl. Prop.) 5. (AAS) m LMP m JLK m JKL MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle. a. GEOMETRIC Use a ruler and a protractor to esolutions Manual - Powered by Cognero Page 8

9 draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown. b. TABULAR Use a protractor to measure and record m 1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables. c. VERBAL Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. d. ALGEBRAIC If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures. c. 5 is supplementary to 1, so m 5 = 180 m , so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 m is supplementary to 3, so m 3 = 180 m 2. m 2 is twice as much as m 4 and m 5, so m 4 = m 5 = a. d. m 5 = 180 x, m 4 = 180 x, m 3 = 2x 180; m 3 = 180 x, m 4 = m 5 = b. esolutions Manual - Powered by Cognero Page 9

10 45. JUSTIFY ARGUMENTS In the figure, if is equilateral and ZWP WJM JZL, prove that Given: is equilateral, and ZWP WJM JZL. Prove: We know that is equilateral, since an equilateral is equiangular, ZWJ WJZ JZW. So, m ZWJ = m WJZ = m JZW, by the definition of congruence. Since ZWP WJM JZL, m ZWP = m WJM = m JZL, by the definition of congruence. By the Angle Addition Postulate, m ZWJ = m ZWP + m PWJ, m WJZ = m WJM + m MJZ, m JZW = m JZL + m LZW. By substitution, m ZWP + m PWJ = m WJM + m MJZ = m JZL + m LZW. Again by substitution, m ZWP + m PWJ = m ZWP + m PJZ = m ZWP + m LZW. By the Subtraction Property, m PWJ = m PJZ = m LZW. By the definition of congruence, PWJ PJZ LZW. So, by ASA, By CPCTC, ANALYZE RELATIONSHIPS Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. Sometimes; only if the measure of the vertex angle is even. 47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. Never; the measure of the vertex angle will be 180 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will be even and 180 2(measure of the base angle) will be even. 48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. Neither; m G = or ORGANIZE IDEAS If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. It is not possible because a triangle cannot have more than one obtuse angle. esolutions Manual - Powered by Cognero Page 10

11 50. ORGANIZE IDEAS In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. 54. QRV is an equilateral triangle. 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? Sample answer: If a triangle is already classified, you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof. What additional information would be enough to prove VRT is an equilateral triangle? F G m VRT = 60 H QRV VTU J VRT RST H 52. ACT/SAT What is m FEG? A 34º B 56 C 60 D 62 E 124 D 53. GRIDDABLE In the figure below, NDG LGD. What is the value of x? 54 esolutions Manual - Powered by Cognero Page 11

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