Lesson 3.2: Triangle Congruence Criteria

When Are Two Triangles the Same?

Apply SAS, ASA, SSS, AAS, and HL to prove triangles congruent. Understand each condition and avoid common misconceptions.

Learning Objectives

State and use SAS, ASA, SSS, AAS
Apply HL in right triangles
Distinguish valid vs. invalid criteria (e.g., SSA)
Explain congruence via rigid motions

Core Criteria

SAS (Side-Angle-Side)

Two sides and the included angle are equal

ASA (Angle-Side-Angle)

Two angles and the included side are equal

SSS (Side-Side-Side)

All three corresponding sides are equal

AAS (Angle-Angle-Side)

Two angles and a non-included side are equal

HL (Hypotenuse-Leg) for Right Triangles

Right triangles with equal hypotenuse and one leg are congruent

Invalid Case: SSA (Side-Side-Angle)

SSA generally does not determine a unique triangle (ambiguous case). Two non-congruent triangles can satisfy SSA with the given data.

Advanced Worked Examples

A) Rigid-Motion Perspective

Show SAS implies a rigid motion mapping one triangle to another.

Translate a vertex to coincide, rotate to align included angle, and reflect if needed to match orientation; equal included sides fix position → triangles coincide.

B) SSA Ambiguity

Given side a, side b adjacent to angle $alpha$ opposite a, with $a<b$ : two distinct triangles may exist if $0<a<bsinalpha$ vs $bsinalpha<a<b$ (none/one/two solutions).

Use the Law of Sines setup to illustrate multiple possible heights to side b.

C) HL Conditions

HL requires right triangles (a 90° angle), equal hypotenuse, and one equal leg. Without a right angle, HL does not apply.

Common Pitfalls

Using SSA as if it guarantees congruence (it does not, except special cases).
Confusing included vs. non-included angles (SAS vs. AAS matters).
Applying HL without confirming both triangles are right triangles.

Worked Example

$AB=DE$ , $\angle A=\angle D$ , $AC=DF$ . Decide if △ABC ≅ △DEF and justify.

Yes, by SAS: two sides and included angle are equal → triangles are congruent.

Practice

1) Which criterion fits: AB=DE, BC=EF, AC=DF?

Solution

SSS

2) Right triangles with hypotenuse and one leg equal?

Solution

3) Give a counterexample to SSA guaranteeing congruence.

Hint/Solution

Fix two sides and a non-included angle; reflect possible height across the base to get two non-congruent configurations with same SSA data.

← Previous Lesson Next Lesson: Proving Congruence →