Triangle Congruence Tests

An interactive demonstration of the four main tests for triangle congruence: SSS, SAS, ASA, and AAS.

Keywords: congruence, SSS, SAS, ASA, AAS, triangle congruence tests

Difficulty: beginner

Two triangles are congruent when they have exactly the same size and shape, written $△ABC\cong △DEF$. But how few measurements must match before congruence is guaranteed?

Strategy: we walk through the four tests that suffice — $SSS$, $SAS$, $ASA$, and $AAS$ — and then show why $SSA$ and $AAA$ do not.

What are Congruent Triangles?

Congruent triangles are identical — they have the same size and shape. Symbol: $△ABC\cong △DEF$ means "triangle ABC is congruent to triangle DEF."

Test 1: SSS (Side-Side-Side)

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

Test 2: SAS (Side-Angle-Side)

If two sides and the INCLUDED angle (between them) are equal, the triangles are congruent. The angle must be BETWEEN the two sides!

Test 3: ASA (Angle-Side-Angle)

If two angles and the INCLUDED side (between them) are equal, the triangles are congruent. The side must be BETWEEN the two angles!

Test 4: AAS (Angle-Angle-Side)

If two angles and a NON-INCLUDED side are equal, the triangles are congruent. This works because the third angle is automatically determined!

What Doesn't Work: SSA

SSA (Side-Side-Angle) is ambiguous — two different triangles can have the same two sides and the same non-included angle!

What Doesn't Work: AAA

AAA (Angle-Angle-Angle) only proves similarity, not congruence — triangles can have the same angles but different sizes.

Summary - The Four Congruence Tests

The four congruence tests:

• SSS: All three sides equal
• SAS: Two sides and included angle equal
• ASA: Two angles and included side equal
• AAS: Two angles and any side equal

Remember: SSA and AAA do NOT prove congruence!

Triangle congruence means two triangles are exactly the same — same size, same shape. You can prove congruence using:

• SSS: Three sides equal
• SAS: Two sides and the included angle equal
• ASA: Two angles and the included side equal
• AAS: Two angles and any side equal

Remember: SSA doesn't work (it's ambiguous), and AAA only proves similarity, not congruence!

Notes

Two triangles are congruent if they have exactly the same size and shape: every pair of corresponding sides is equal in length and every pair of corresponding angles is equal in measure, so one triangle is an exact copy of the other (possibly rotated or flipped). In symbols, $△ABC\cong △DEF$.

The Four Congruence Tests

1. SSS (Side-Side-Side)

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

2. SAS (Side-Angle-Side)

If two sides and the included angle (the angle between those two sides) are equal, the triangles are congruent. The angle must be BETWEEN the two sides!

3. ASA (Angle-Side-Angle)

If two angles and the included side (the side between those two angles) are equal, the triangles are congruent. The side must be BETWEEN the two angles!

4. AAS (Angle-Angle-Side)

If two angles and any side are equal, the triangles are congruent. This works because once two angles are fixed, the third angle is determined (the angles sum to $180°$).

What Doesn't Work?

• SSA (Side-Side-Angle): ambiguous — two different triangles can satisfy the same SSA conditions.
• AAA (Angle-Angle-Angle): only proves similarity, not congruence — triangles can have the same angles but different sizes.