4.6 Isosceles Triangles

4.6 Isosceles TrianglesTheorem 4.9 Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the angles opposite those sides are congruent.Theorem 4.10If two angles of a triangle are congruent, then the sides opposite those angles are congruent.Corollaries4.3 A triangle is equilateral if and only if it is equiangular4.4 Each angle of an equilateral triangle measures 60

Example 6-1b

Proof: ReasonsStatements3. Def. of isosceles 3. ABC and BCD are isosceles1. Given1.

6.6. Substitution

5.5. Given

4.4. Isosceles Theorem

2. Def. of segments2.

Example 6-1c

Write a two-column proof.Given: . Prove:

Example 6-1d

Proof: ReasonsStatements1. Given3. Isosceles Theorem 2. Def. of isosceles triangles1.

2. ADB is isosceles.3.4.5.4. Given5. Def. of midpoint6. SAS7.7. CPCTC

6. ABC ADC

Example 6-2a

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. 75Example 6-2b

Solve the Test Item

Angle Sum TheoremSubstitutionAdd.Subtract 120 from each side.Divide each side by 2.Step 1 The base angles of CDE are congruent. Let

Example 6-2c

Def. of vertical anglesSubstitutionAdd.Step 2 are vertical angles so they have equal measures.

Step 3 The base angles of CBA are congruent.Angle Sum TheoremSubstitutionSubtract 30 from each side.Divide each side by 2.

Example 6-2d

Answer: DExample 6-2e

Answer: A

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

A. 25 B. 35 C. 50 D. 130Example 6-3a

Answer:

Name two congruent angles.

Example 6-3b

Answer:

Name two congruent segments.

By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,Example 6-3c

a. Name two congruent angles. Answer:Answer:b. Name two congruent segments.

Example 6-4a

Since the angle was bisected,

Each angle of an equilateral triangle measures 60.

EFG is equilateral, and bisects bisectsFind and

Example 6-4b

Answer:Add.Exterior Angle TheoremSubstitution

is an exterior angle of EGJ.

Example 6-4c

Subtract 75 from each side.Linear pairs are supplementary.Substitution

Answer: 105

EFG is equilateral, and bisects bisectsFind

a. Find x. Example 6-4d

b. Answer: 90Answer: 30

ABC is an equilateral triangle. bisects