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4-6 Triangle Congruence: ASA, AAS, and HL
Learning Targets
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.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Vocabulary
included side
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: KNL  MLN , KL || MN
Prove: NKL  LMN.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Statements
Reasons
1. KNL  MLN , KL || MN 1. Given
2. Alternate Interior Angles Theorem
2. KLN  MNL
3. NL  NL
3. Reflexive Property of
4. NKL  LMN
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  M
Prove: 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
1. JL bisects KLM , K  M
2. KLJ  MLJ
Reasons
1. Given
3. JL  JL
2. Def. Angle Bisector
3. Reflexive Property of 
4. JKL  JML
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
Holt McDougal Geometry