The Four Congruence Conditions for Triangles

The Four Congruence Conditions for Triangles

You only need to check a minimum set of measurements to guarantee two triangles are congruent. There are exactly four valid conditions:

1. SSS — Side-Side-Side

If all three sides of one triangle equal the three sides of another (in corresponding pairs), the triangles are congruent.

Triangle ABC ≅ Triangle DEF if: AB = DE, BC = EF, AC = DF

Why it works: Fixing three side lengths determines the triangle completely — there is only one triangle (up to reflection) with those three side lengths.

2. SAS — Side-Angle-Side

If two sides and the included angle (the angle between those two sides) are equal, the triangles are congruent.

Triangle ABC ≅ Triangle DEF if: AB = DE, angle B = angle E, BC = EF

Critical detail: The angle MUST be included (between) the two sides. Knowing two sides and a non-included angle is NOT sufficient (see SSA below).

3. ASA — Angle-Side-Angle

If two angles and the included side (the side between those two angles) are equal, the triangles are congruent.

Triangle ABC ≅ Triangle DEF if: angle A = angle D, AB = DE, angle B = angle E

Note: AAS (Angle-Angle-Side where the side is NOT included) is also valid because if two angles are known, the third is determined (angles sum to 180°), so AAS effectively gives you ASA.

4. RHS — Right angle-Hypotenuse-Side

For right-angled triangles only: if the hypotenuse and one other side are equal, the triangles are congruent.

Triangle ABC ≅ Triangle DEF if: angle C = angle F = 90°, AB = DE (hypotenuse), AC = DF (or BC = EF)

Why it works: The third side is determined by Pythagoras' theorem once two sides are fixed in a right-angled triangle.