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 swinging side can land in two places. AAA proves triangles are the same shape (similar),
not the same size. Neither proves congruence.
triangles congruent; CPCTC is the reason you then claim a side or angle is congruent. Always finish with it.
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.