Section 4-4 Isosceles Triangles & Proofs

1. Section 4-4 Isosceles Triangles & Proofs

2. Review Definition of Isosceles Triangle: A triangle with at least two sides congruent. Theorem 4-1: If a triangle has two congruent sides, the angles opposite those sides are congruent. B So, A  C A C

3. PROOF OF THEOREM 4-1: B Given: BD bisects ABC AB  BC Prove: A  C A D C Statements Reasons 1. BD bisects ABC 1. Given 2. ABD  CBD 2. Def. of Angle Bisector 3. AB  BC 3. Given 4. BD  BD 4. Reflexive Property 5. ΔABD ΔCBD 5. SAS Postulate 6. A  C 6. CPCTC

4. Theorem 4-2: If two angles of a triangle are congruent, the sides opposite those angles are congruent. B C A So, AB  BC

5. PROOF EXAMPLE 1: X Given: XY  XZ Prove: 1 3 1 2 Y Z 3 Statements Reasons 1. XY  XZ 1. Given 2. If two sides of a triangle are congruent, the angles opposite those sides are congruent. 2. 1  2 3. 2  3 3. Vertical Angle Theorem 4. 1  3 4. Substitution

6. PROOF EXAMPLE 2: R Given: RS  RT Prove: 3 4 S T 1 2 3 4 Statements Reasons 1. RS  RT 1. Given 2. If two sides of a triangle are congruent, the angles opposite those sides are congruent. 2. 1 2 3. 1 3, 2 4 3. Vertical Angles Theorem 4. 2 3 4. Substitution 5. 3 4 5. Substitution

7. PROOF EXAMPLE 3: Given: XY  XZ OY  OZ Prove: m1= m4 X O 1 4 2 3 Y Z

8. Statements Reasons 1. XY  XZ 1. Given 2. If two sides of a triangle are congruent, the angles opposite those sides are congruent. • XYZ XZY • mXYZ= mXZY 3. OY  OZ 3. Given 4. If two sides of a triangle are congruent, the angles opposite those sides are congruent. 4. 2 3; m2= m3 • m1+ m2 = mXYZ • m3+ m4 = mXZY 5. Angle Addition Postulate 6. m1+ m2 = m3+ m4 6. Substitution 7. m1 = m4 7. Subtraction