Proving Triangles Congruent: ASA, AAS, and HL
State and apply the ASA, AAS congruence theorems and the HL theorem for right triangles. Determine which congruence shortcut applies to a given pair of triangles and write complete proofs.
A lesson of Geometry, in Triangles and Congruence, about 10 minutes.
By the end of this lesson, you will be able to apply the ASA and AAS congruence theorems and the HL theorem for right triangles, distinguish between valid and invalid shortcuts (including AAA and SSA), and write proofs using all five congruence methods.
The Big Idea
We now have five total congruence methods: SSS, SAS, ASA, AAS, and HL. Together they cover every valid shortcut. The two invalid ones — AAA and SSA — are equally important to know, because using them is a common proof error.
ASA — Angle-Side-Angle
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.
The included side is between the two angles.
$$\angle A \cong \angle D, \quad \overline{AB} \cong \overline{DE}, \quad \angle B \cong \angle E \implies \triangle ABC \cong \triangle DEF$$
AAS — Angle-Angle-Side
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.
AAS is a theorem (can be proved from ASA), not a postulate.
$$\angle A \cong \angle D, \quad \angle B \cong \angle E, \quad \overline{BC} \cong \overline{EF} \implies \triangle ABC \cong \triangle DEF$$
Note: AAS vs. ASA: in AAS, the side is NOT between the two angles. The side still corresponds — it must be the side opposite one of the corresponding angles.
HL — Hypotenuse-Leg (Right Triangles Only)
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.
Conditions: both triangles must be right triangles, the hypotenuses must be congruent, and one pair of legs must be congruent.
HL is a theorem that follows from SSS (given right angles and two sides, the third side is determined by the Pythagorean theorem).
Invalid Shortcuts
AAA (Angle-Angle-Angle): Three congruent angle pairs → triangles are similar, NOT necessarily congruent. They could be the same shape but different sizes.
SSA (Side-Side-Angle): The ambiguous case — two triangles can share two sides and a non-included angle but be different triangles.
Example 1 — Identifying the Method
(a) $\angle A \cong \angle D$, $\overline{AC} \cong \overline{DF}$, $\angle C \cong \angle F$ — side is between the two angles → ASA
(b) $\angle P \cong \angle X$, $\angle Q \cong \angle Y$, $\overline{QR} \cong \overline{YZ}$ — side is NOT the included side →