Warm-up

AASSSS

Not possible

HL

Not possible SAS

4.6 Isosceles, Equilateral, and Right Triangles

Students will use the Isosceles Base Angles Theorem and the HL

theorem to prove triangles congruent.

Given:

Prove: B C

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ΔABC, AB≅AC, D is the midpoint ofCB.

A

BC D

Base Angles Theorem

• If two sides of a triangle are congruent, then the angles opposite them are congruent.

• If , then B C.A

B C

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AB≅AC

Converse of the Base Angles Theorem

• If two angles of a triangle are congruent, then the sides opposite them are congruent.

• If B C, then .

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AB≅AC

Corollary to the Base Angles Theorem

• If a triangle is equilateral, then it is equiangular.

Corollary to the Converse of the Base Angles Theorem

• If a triangle is equiangular, then it is equilateral.

Hypotenuse-Leg (HL) Congruence Theorem

• If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

• If and , then ΔABC ΔDEF. A

B C

D

E F

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AC≅DF

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BC≅EF

Example Proof:

• The television antenna is to the plane containing the points B, C, D, and E. Each of the stays running from the top of the antenna to B, C, and D uses the same length of cable. Prove that ΔAEB, ΔAEC, and ΔAED are congruent.

• Given:

• Prove: ΔAEB ΔAEC ΔAED

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AE⊥EB, AE⊥EC, AE⊥ED, AB≅AC≅AD

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