4.6 Isosceles Triangles4.6 Isosceles Triangles

ObjectivesObjectives

Use properties of isosceles trianglesUse properties of isosceles triangles

Use properties of equilateral trianglesUse properties of equilateral triangles

Properties of Isosceles TrianglesProperties of Isosceles Triangles

The The formed by the ≅ sides is formed by the ≅ sides is called the called the vertex anglevertex angle. .

The two ≅ sides are called The two ≅ sides are called legslegs. The third side is called . The third side is called the the basebase..

The two The two s formed by the base s formed by the base and the legs are called theand the legs are called thebase anglesbase angles. .

leg leg

base

vertex

Isosceles Triangle TheoremIsosceles Triangle Theorem

Theorem 4.9Theorem 4.9If two sides of a ∆ are If two sides of a ∆ are ≅, then the ≅, then the s s opposite those sides are ≅ (if AC ≅ AB, opposite those sides are ≅ (if AC ≅ AB, then then B ≅ B ≅ C).C). A

B C

Write a two-column proof.

Given:

Prove:

Example 1:Example 1:

Proof:

ReasonsReasonsStatementsStatements

3.3. Def. of Isosceles Def. of Isosceles 33. . ABCABC and and BCDBCD are are isosceles trianglesisosceles triangles

1.1. GivenGiven1.1.

6.6. 6.6. SubstitutionSubstitution

5.5. 5.5. GivenGiven

4.4. 4.4. Isosceles Isosceles Theorem Theorem

2.2. Def. of SegmentsDef. of Segments2.2.

Example 1:Example 1:

Write a two-column proof.

Given: .

Prove:

Your Turn:Your Turn:

Proof:

ReasonsReasonsStatementsStatements

1.1. GivenGiven

3.3. Isosceles Isosceles Theorem Theorem

2.2. Def. of Isosceles TrianglesDef. of Isosceles Triangles

1.1.

2.2. ADBADB is isosceles. is isosceles.

3.3.

4.4.

5.5.

4.4. GivenGiven

55. . Def. of MidpointDef. of Midpoint

6.6. SASSAS

7.7. 7.7. CPCTCCPCTC66. . ABCABC ADCADC

Your Turn:Your Turn:

The Converse of Isosceles The Converse of Isosceles Triangle TheoremTriangle Theorem

Theorem 4.10Theorem 4.10

If two If two ss of a ∆ are of a ∆ are ≅, then the sides ≅, then the sides opposite those opposite those s are ≅ (if s are ≅ (if B ≅ B ≅ C, C, then AC ≅ AB).then AC ≅ AB).

Answer:

Name two congruent angles.

Example 2:Example 2:

Answer:

Name two congruent segments.

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

Example 2:Example 2:

a. Name two congruent angles.

Answer:

Answer:

b. Name two congruent segments.

Your Turn:Your Turn:

Properties of Equilateral ∆sProperties of Equilateral ∆s

Corollary 4.3Corollary 4.3A ∆ is equilateral iff it is equiangular.A ∆ is equilateral iff it is equiangular.

Corollary 4.4Corollary 4.4Each Each of an equilateral of an equilateral ∆∆ measures measures 6060°.°.

Since the angle was bisected,Each angle of an equilateral triangle measures 60°.

EFG is equilateral, and bisects bisectsFind and

Example 3a:Example 3a:

Answer:

Add.

Exterior Angle Theorem

Substitution

is an exterior angle of EGJ.

Example 3a:Example 3a:

Subtract 75 from each side.

Linear pairs are supplementary.

Substitution

Answer: 105

EFG is equilateral, and bisects bisectsFind

Example 3b:Example 3b:

a. Find x.

b.

Answer: 90

Answer: 30

ABC is an equilateral triangle. bisects

Your Turn:Your Turn: