4.6 Isosceles Triangles

• Theorem 4.9 Isosceles Triangle Theorem– If two sides of a triangle are congruent, then the

angles opposite those sides are congruent.• Theorem 4.10– If two angles of a triangle are congruent, then the

sides opposite those angles are congruent.• Corollaries– 4.3 A triangle is equilateral if and only if it is

equiangular– 4.4 Each angle of an equilateral triangle measures 60°

Write a two-column proof.

Given:

Prove:

Proof:

ReasonsStatements

3. Def. of isosceles 3. ABC and BCD are isosceles

1. Given1.

6. 6. Substitution

5. 5. Given

4. 4. Isosceles Theorem

2. Def. of segments2.

Write a two-column proof.

Given: .

Prove:

Proof:

ReasonsStatements

1. Given

3. Isosceles Theorem

2. Def. of isosceles triangles

1.

2. ADB is isosceles.

3.

4.

5.

4. Given

5. Def. of midpoint

6. SAS

7. 7. CPCTC

6. ABC ADC

Multiple-Choice Test Item If and what is the measure of

Read the Test Item CDE is isosceles with base Likewise, CBA is isosceles with

A. 45.5 B. 57.5 C. 68.5 D. 75

Solve the Test Item

Angle Sum Theorem

Substitution

Add.

Subtract 120 from each side.

Divide each side by 2.

Step 1 The base angles of CDE are congruent. Let

Def. of vertical angles

Substitution

Add.

Step 2 are vertical angles so they have equal measures.

Step 3 The base angles of CBA are congruent.

Angle Sum Theorem

Substitution

Subtract 30 from each side.

Divide each side by 2.

Answer: D

Answer: A

Multiple-Choice Test Item If and what is the measure of

A. 25 B. 35 C. 50 D. 130

Answer:

Name two congruent angles.

Answer:

Name two congruent segments.

By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,

a. Name two congruent angles.

Answer:

Answer:

b. Name two congruent segments.

Since the angle was bisected,

Each angle of an equilateral triangle measures 60°.

EFG is equilateral, and bisects bisectsFind and

Answer:

Add.

Exterior Angle Theorem

Substitution

is an exterior angle of EGJ.

Subtract 75 from each side.

Linear pairs are supplementary.

Substitution

Answer: 105

EFG is equilateral, and bisects bisectsFind

a. Find x.

b.

Answer: 90

Answer: 30

ABC is an equilateral triangle. bisects