DWG NO. 04.4 — Lesson 4 of 6

Proving Congruence: ASA, AAS, and HL

Unit 4: Congruent Triangles · ~15–30 min

Objective

Use the ASA, AAS, and HL congruence shortcuts to prove two triangles are congruent, and recognize when SSA doesn't work.

Three more shortcuts

SSS and SAS aren't the only ways to lock a triangle's shape with just three known parts. Two more general shortcuts use mostly angles, and one special shortcut applies only to right triangles.

Shortcut	What's given	Key detail
ASA	Two angles and the included side	The side must be between the two given angles.
AAS	Two angles and a non-included side	The side is next to one angle but not between them.
HL	Hypotenuse and one leg (right triangles only)	Only valid when both triangles are already known to be right triangles.

ASA Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another, the triangles are congruent.
AAS Theorem: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another, the triangles are congruent.
HL Theorem: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent.

Why SSA doesn't make the list

Two sides and a non-included angle (SSA) is not a valid congruence shortcut in general — the same two sides and angle can sometimes be hinged into two different triangles. It's often nicknamed the "donkey theorem" for this reason. HL is really just a special case where SSA does work, because the 90° angle and the Pythagorean relationship remove the ambiguity.

Two right triangles with matching hypotenuse (BC and EF) and one leg (AB and DE) — congruent by HL.

Worked Example 1 · Identify ASA vs. AAS

ProblemTriangle RST: angle R = 40°, angle S = 65°, side RT = 8. Triangle UVW: angle U = 40°, angle V = 65°, side UW = 8. Side RT lies opposite angle S, and side UW lies opposite angle V — neither side sits between the two given angles. Which postulate applies?

1.Two angles are given in each triangle (40° and 65°), plus one side. Since that side is not between the two given angles, it's a non-included side.

Triangle RST ≅ triangle UVW by AAS

Worked Example 2 · Apply HL

ProblemRight triangles JKL and MNP each have a right angle at K and N. Given hypotenuse JL = MP = 13 and leg JK = MN = 5, prove the triangles congruent.

1.Both triangles are confirmed right triangles (right angles at K and N), which is required before HL can apply.

2.The hypotenuses match (JL = MP = 13) and one pair of legs matches (JK = MN = 5).

Triangle JKL ≅ triangle MNP by HL

Guided practice

1.Two triangles share two congruent angles and the side between them. Name the postulate.
Show answer
ASA — the side is included between the two angles.
2.Two triangles share two congruent angles and a side that is not between them. Name the theorem.
Show answer
AAS.
3.Explain why "SSA" is not a valid triangle congruence shortcut in general.
Show answer
Given two sides and a non-included angle, the third side can sometimes swing to two different lengths and still satisfy those same measurements — producing two non-congruent triangles from the same SSA data. So SSA doesn't guarantee a single triangle shape.
4.Two right triangles have congruent hypotenuses but no other parts confirmed congruent. Is that enough for HL?
Show answer
No — HL needs the hypotenuse and one leg to be congruent. A matching hypotenuse alone isn't enough.
5.Triangle ABC and triangle DEF each have a right angle at B and E. AB = DE and angle A = angle D, where AB is a leg (not the hypotenuse) and angle A is not the right angle. Which shortcut proves congruence — HL, or one of the angle-based ones?
Show answer
Since the right angles at B and E are congruent, and angle A ≅ angle D is given, plus leg AB ≅ DE is included between angle A and angle B — this is ASA (two angles and the included side), not HL, since HL specifically needs the hypotenuse.

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