Review

Definition of Isosceles Triangle:

Theorem 4-1: If a triangle has two congruent sides, the angles opposite those

sides are congruent.

A triangle with at least two sides congruent.

B

A C

So, A C

PROOF OF THEOREM 4-1:

Given: BD bisects ABC

AB BC

Prove: A C

1. Given

4. Reflexive Property

6. CPCTC6. A C

5. ΔABD ΔCBD

2. ABD CBD 2. Def. of Angle Bisector

5. SAS Postulate

B

DA C

3. Given

Statements Reasons

1. BD bisects ABC

3. AB BC

4. BD BD

Theorem 4-2: If two angles of a triangle are congruent, the sides

opposite those angles are congruent.

B

A C

So, AB BC

X

ZY 321

PROOF EXAMPLE 1:

Given: XY XZ

Prove: 1 3

1. XY XZ 1. Given

4. 1 3 4. Substitution

2. 1 2

2. If two sides of a triangle are congruent, the angles opposite those sides are congruent.3. 2 3 3. Vertical Angle Theorem

Statements Reasons

R

TS

321

4

PROOF EXAMPLE 2:

Given: RS RT

Prove: 3 4

1. RS RT 1. Given

4. 2 3 4. Substitution

5. Substitution 5. 3 4

2. 1 2

3. 1 3, 2 4 3. Vertical Angles Theorem

Statements Reasons

2. If two sides of a triangle are congruent, the angles opposite those sides are congruent.

PROOF EXAMPLE 3:

32

1 4

X

ZY

O

Given: XY XZ

OY OZ

Prove: m1 = m4

1. XY XZ 1. Given

4. 2 3; m2 = m3

7. Subtraction

2. XYZ XZY

mXYZ = mXZY

6. Substitution

3. OY OZ 3. Given

Statements Reasons

5. m1 + m2 = mXYZ

m3+ m4 = mXZY

6. m1 + m2 = m3+ m4

7. m1 = m4

5. Angle Addition Postulate

2. If two sides of a triangle are congruent, the angles opposite those sides are congruent.

4. If two sides of a triangle are congruent, the angles opposite those sides are congruent.