Properties of Isosceles and Equilateral Triangles


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1 Properties of Isosceles and Equilateral Triangles
2 In an isosceles triangle, the sides and the angles of the triangle are classified by their position in relation to the triangle s congruent sides. Leg of an Isosceles Triangle  One of the two congruent sides of the triangle. In the diagram, AB and AC are the legs. Vertex Angle of an Isosceles Triangle  The angle formed by the legs of the triangle. The vertex angle is A. Base of an Isosceles Triangle  The side opposite the vertex angle. The base of ABC is BC. Base Angle of an Isosceles Triangle  One of the two angles that have the base of the triangle as a side. In ABC, B and C are base angles.
3 Theorem 511: Isosceles Triangle Theorem  If a triangle is isosceles, then its base angles are congruent. LMN is isosceles. Therefore, M N. Corollary If a triangle is equilateral, then it is equiangular.
4 Prove the Isosceles Triangle Theorem. Given: ABC is an isosceles triangle with AB AC. D is the midpoint of BC. Prove: B C SOLUTION Statements Reasons 1. ABC is isosceles 1. Given 2. AB AC 2. Definition of isosceles triangle 3. BD = CD 3. Definition of midpoint 4. BD CD 4. Definition of congruent segments 5. AD AD 5. Reflexive Property 6. ABD ACD 6. SSS Triangle Congruence Postulate 7. B C 7. CPCTC
5 Theorem 512: Converse of the Isosceles Triangle Theorem  If two angles of a triangle are congruent, then the sides opposite those angles are also congruent. Corollary If a triangle is equiangular, then it is equilateral.
6 a. Triangle DEF is isosceles, and its vertex angle is at E. If m D = 36, determine m E and m F. SOLUTION The base angles of DEF D and E, so by the Isosceles Triangle Theorem, D F. By the definition of congruent angles, m F = m D, so they each measure 36. Therefore, m D + m E + m F = 180 TAST 36 + m E + 36 = 180 Substitute m E = 108 Solve
7 b. The perimeter of GHJ is 12 inches, and G H. If GH = 5 inches, find GJ. SOLUTION By the Converse of the Isosceles Triangle Theorem, GJ HJ. Since the perimeter is 8 inches and GH = 5 inches, P = GH + HJ + GJ Formula for perimeter 12 = 5 + HJ + GJ Substitute. 12 = 5 + GJ + GJ Def of cong segments 12 = 5 + 2GJ Simplify. GJ = 3.5 in. Solve.
8 A triangle is equiangular and has a perimeter of 22.5 centimeters. Determine the length of each side. SOLUTION By Corollary , the triangle is equilateral. Let the length of each side be s. The perimeter is the sum of the three sides. P = s + s + s Formula for perimeter 22.5 = 3s Substitute and simplify. s = 7.5 cm Solve.
9 Theorem If a line bisects the vertex angle of an isosceles triangle, then it is the perpendicular bisector of the base. Theorem If a line is the perpendicular bisector of the base of an isosceles triangle, then it bisects the vertex angle. The diagram illustrates both of these theorems. The altitude TU bisects the vertex angle and is a perpendicular bisector of the base of the triangle.
10 a. Prove Theorem Given: ABC is isosceles, AD bisects A Prove: AD is the perpendicular bisector of BC SOLUTION Statements Reasons 1. ABC is isosceles, AD bisects A 1. Given 2. BAD CAD 2. Definition of angle bisector 3. AB AC 3. Definition of isosceles triangle 4. AD AD 4. Reflexive Property 5. ABD ACD 5. SAS Triangle Congruence Postulate 6.BD CD 6. CPCTC 7. BD = CD 7. Definition of congruent segments 8. ADB ADC 8. CPCTC 9. AD and BC form adjacent angles 9. Definition of adjacent angles 10. AD BC 10. If lines form congruent adjacent angles, they are perpendicular 11. AD is bisector of BC 11. Def of perpendicular bisector
11 b. Write a paragraph proof of Theorem Given: ABC is isosceles, AD is the perpendicular bisector of BC Prove: AD bisects A SOLUTION Since ABC is isosceles, AB AC. By the Reflexive Property, AD AD. Both ABD and ACD are right triangles, since AD is the perpendicular bisector of BC and forms two right angles at D. Therefore, ABD ACD by the HypotenuseLeg Right Triangle Congruence Theorem. By CPCTC, BAD CAD. Therefore, by the definition of an angle bisector, AD bisects BAC.
12 This figure shows the north and east view of a telephone pole that is secured by four cables of equal length. a. Explain why the base angles, PAQ and PRQ, are congruent. SOLUTION In APR, the cable lengths AP and RP are equal, so AP RP by the definition of congruent segments. Therefore, APR is isosceles by definition. Applying the Isosceles Triangle Theorem, the base angles of APR are congruent, so PAQ PRQ.
13 This figure shows the north and east view of a telephone pole that is secured by four cables of equal length. b. Prove that these angles are also congruent to the base angles B and D. SOLUTION By the Reflexive Property of Congruence, PQ PQ. It is given in the problem that BP AP, so by the HypotenuseLeg Right Triangle Congruence Theorem, BPQ APQ. By CPCTC, B A. Since BPD is isosceles, D B by the Isosceles Triangle Theorem. It is given that A R, so by the Transitive Property of Congruence, R D.
14 a.for the isosceles triangle shown, determine the missing angle measures. b. The perimeter of XYZ is 15.2 centimeters, and X Z. If XY = 6.3 centimeters, determine XZ.
15 c. If the vertex angle of an isosceles triangle measures 20, what are the measures of each of its base angles? d. A triangle is equiangular and its perimeter is 7 feet. Determine the length of each side.
16 e.engineering: This diagram shows the sideview profile of a bridge. Determine the angle that each half of the bridge makes with the horizontal.
17 Page 339 Lesson Practice (Ask Mr. Heintz) Page 340 Practice 130 (Do the starred ones first)
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