Triangle Congruence (SSS, SAS, ASA, AAS)

Geometry · Axiom Academy

LESSON Triangle Congruence (SSS, SAS, ASA, AAS) SSS, SAS, ASA, and AAS — proving two triangles are identical. 1. What Does "Congruent" Mean? Two triangles are congruent if you can slide and turn one so it lands exactly on top of the other. That means all corresponding sides are equal and all corresponding angles are equal — but a rigid motion (sliding + rotating, no stretching) is what makes "same size and shape" precise. means all three sides AND all three angles match 2. The Four Congruence Postulates Each of these four combinations is enough to force two triangles to be congruent. Tick marks show equal sides; arcs show equal angles. If all three sides of one triangle equal all three sides of another, the triangles are congruent. If two sides and the included angle (the angle between them) are equal, the triangles are congruent. If two angles and the included side (the side between them) are equal, the triangles are congruent. If two angles and a non-included side are equal, the triangles are congruent (the third angle is forced, so this reduces to ASA). Three equal angles only prove the triangles are similar , not congruent — they can be different sizes (think two equilateral triangles with different side lengths). Two sides and a non-included angle can build two different triangles — the "ambiguous case." We watch exactly that happen in the next step. 3. Quick Reference & Why SSA Fails

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