Triangle Congruence Theorems

Congruent Triangles

$\triangle ABC \cong \triangle DEF$ means all corresponding sides AND angles are equal.

The Five Congruence Criteria

SSS (Side-Side-Side)

If all three sides of one triangle are congruent to all three sides of another, the triangles are congruent.

SAS (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to the corresponding parts of another.

ASA (Angle-Side-Angle)

If two angles and the included side of one triangle are congruent to the corresponding parts of another.

AAS (Angle-Angle-Side)

If two angles and a non-included side of one triangle are congruent to the corresponding parts of another.

HL (Hypotenuse-Leg)

For right triangles only: If the hypotenuse and one leg are congruent.

What Does NOT Work

• SSA (Side-Side-Angle): Ambiguous case — can produce two different triangles
• AAA (Angle-Angle-Angle): Only proves similarity, not congruence

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Once you prove $\triangle ABC \cong \triangle DEF$, you can conclude:

• $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\overline{AC} \cong \overline{DF}$
• $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$

Writing a Congruence Proof

1. Given: State what information you have
2. Identify shared parts: Look for shared sides, vertical angles, or parallel lines
3. Choose a theorem: SSS, SAS, ASA, AAS, or HL
4. State the congruence: $\triangle ABC \cong \triangle XYZ$
5. CPCTC: Use to prove additional parts equal

Order matters! $\triangle ABC \cong \triangle DEF$ means $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$.