Skill: congruent-triangles

Congruent Triangles

Five shortcuts prove triangles congruent — SSS, SAS, ASA, AAS, HL. Two famous fakes don’t: SSA and AAA. Then CPCTC finishes the job.

1 The five shortcuts

Each one is a minimum shopping list: if you have exactly these marked pairs, the triangles must be congruent.

SSS Side-Side-Side

All three pairs of sides marked congruent. The tick marks tell the story — no angles needed.

SAS Side-Angle-Side

Two sides and the INCLUDED angle — the angle sitting between the two marked sides.

ASA Angle-Side-Angle

Two angles and the INCLUDED side — the side connecting the two marked angles.

AAS Angle-Angle-Side

Two angles and a NON-included side. The third angle comes free (angle sum), turning AAS into ASA.

HL Hypotenuse-Leg

Right triangles only: the hypotenuse and one leg marked. A shortcut built just for right triangles.

The two fakes: SSA (two sides and a non-included angle) is ambiguous — the swinging side can land in two places. AAA proves triangles are the same shape (similar), not the same size. Neither proves congruence.

CPCTC — “Corresponding Parts of Congruent Triangles are Congruent.” The postulate proves the triangles congruent; CPCTC is the reason you then claim a side or angle is congruent. Always finish with it.

Quick check 1 — All three pairs of corresponding sides are marked congruent. The postulate is…

Quick check 2 — Two right triangles have their hypotenuses and one pair of legs marked. The postulate is…

Quick check 3 — Two pairs of sides and a NON-included angle are marked. Which postulate applies?

Quick check 4 — △ABC ≅ △DEF by SSS. Which reason proves ∠B ≅ ∠E?

2 Worked examples

Two full two-column proofs — the format every congruence problem wants.

Example 1 — SAS proof

Given: E is the midpoint of AC and BD. Prove: △AEB ≅ △CED.

#	Statement	Reason
1	E is the midpoint of AC	Given
2	AE ≅ EC	Definition of midpoint
3	E is the midpoint of BD	Given
4	BE ≅ ED	Definition of midpoint
5	∠AEB ≅ ∠CED	Vertical angles are congruent
6	△AEB ≅ △CED	SAS

The vertical angles sit between the two marked side pairs — that is what makes it SAS, not SSA.

Example 2 — finish with CPCTC

Given: ∠A ≅ ∠D, ∠B ≅ ∠E, BC ≅ EF. Prove: AB ≅ DE.

#	Statement	Reason
1	∠A ≅ ∠D; ∠B ≅ ∠E; BC ≅ EF	Given
2	△ABC ≅ △DEF	AAS
3	AB ≅ DE	CPCTC

The proof is not done at step 2 — step 2 proves the triangles congruent, and only CPCTC transfers that to the parts.

Quick check 5 — △PQR ≅ △STU with P↔S, Q↔T, R↔U. Segment PQ corresponds to…

Quick check 6 — All three pairs of angles are marked congruent, but no sides are marked. Which postulate applies?

Practice congruence proofs Open the postulate identifier