Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

I will apply the ASA Postulate, the AAS Theorem, and the HL Theorem to construct triangles and to solve problems.

I will prove triangles congruent by using the ASA Postulate, AAS Theorem, and HL Theorem.

Learning Targets

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

included side

Vocabulary

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

ASA Postulate

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Example: Applying ASA Congruence

Determine if you can use the ASA Postulate to prove the triangles congruent. Explain.

Two congruent angle pairs are give, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent.

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Check It Out! Example 2

Given:

Prove: NKL LMN.

, || KNL MLN KL MN

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Statements Reasons

1. , ||

2.

3.

4.

KNL MLN KL MN

KLN MNL

NL NL

NKL LMN

1. Given

2. Alternate Interior Angles Theorem

3. Reflexive Property of

4. ASA Postulate

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side Theorem, or AAS Theorem.

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Check It Out! Example 3

Given: JL bisects KLM, K MProve: JKL JML

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Statements Reasons

1. bisects ,

2.

3.

4.

JL KLM K M

KLJ MLJ

JL JL

JKL JML

1. Given

2. Def. Angle Bisector

3. Reflexive Property of

4. AAS Theorem

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Example 4A: Applying HL Theorem

Determine if you can use the HL Theorem to prove the triangles congruent. If not, tell what else you need to know.

According to the diagram, the triangles are right triangles that share one leg. It is given that the hypotenuses are congruent, therefore the triangles are congruent by HL.

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Example 4B: Applying HL Theorem

This conclusion cannot be proved by the HL Theorem. According to the diagram, the triangles are right triangles and one pair of legs is congruent. You do not know that one hypotenuse is congruent to the other.

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Check It Out! Example 4

Determine if you can use the HL Theorem to prove ABC DCB. If not, tell what else you need to know.

Yes; it is given that AC DB. BC CB by the Reflexive Property of Congruence. Since ABC and DCB are right angles, ABC and DCB are right triangles. ABC DCB by HL.

Holt McDougal Geometry

4-6 Triangle Congruence: ASA, AAS, and HL

Homework: Pg 264 – 265, #4 – 17, 19, 22, 23