L1 · Congruence criteria: SSS, SAS, ASA

Learning goal

“I can decide whether two fully described triangles must be congruent by checking SSS, SAS, or ASA against their measures, name the criterion I used, and explain why SSA data can never settle the question.”

Congruent — exactly the same size and shape: every side and every angle of one triangle matches a partner in the other (like two copies from the same stencil).

Corresponding parts — the sides and angles that match up between two triangles; in triangle ABC congruent to triangle DEF, side AB corresponds to side DE.

Included angle — the angle BETWEEN two named sides: in triangle ABC, the angle at B is the included angle of sides AB and BC, because they meet at B.

Included side — the side BETWEEN two named angles: in triangle ABC, side AB is the included side of the angles at A and B, because it joins them.

Worked example

SSS first. Triangle ABC has AB = 5, BC = 7, CA = 6. Triangle DEF has DE = 5, EF = 7, FD = 6. Step 1: Match the sides in order: AB = DE = 5; BC = EF = 7; CA = FD = 6. Three pairs of equal sides. Step 2: Three fixed side lengths can be assembled in only one shape (the third vertex is pinned by the two lengths that must reach it), so the triangles are congruent: triangle ABC is congruent to triangle DEF, by SSS. SAS next. Triangle GHI has GH = 8, angle H = 40 degrees, HI = 11. Triangle JKL has JK = 8, angle K = 40 degrees, KL = 11. Step 1: Two pairs of equal sides: GH = JK = 8 and HI = KL = 11. Step 2: The equal angles sit BETWEEN those sides: angle H is between GH and HI; angle K is between JK and KL. The angles are included angles. Step 3: Two sides and the included angle lock a triangle completely: triangle GHI is congruent to triangle JKL, by SAS. Say it plainly: a criterion is a lock. SSS locks all three sides; SAS locks two sides and the angle wedged between them. If the data locks the triangle into exactly one possible shape, congruence is proven - not suspected.

Check yourself — explain it

For SAS, the equal angle must be the INCLUDED angle because...

Paired problem — example still visible

Triangle MNO has MN = 4, NO = 9, OM = 6. Triangle PQR has PQ = 9, QR = 6, RP = 4. Are the triangles congruent, and by which criterion? -> Yes, by SSS: the side lengths match as 4 = RP, 9 = PQ, 6 = QR (M pairs with R, N with P, O with Q)

Faded completion — last step is yours

"Two triangles whose three pairs of corresponding sides are equal are congruent by the ___ criterion" -> SSS

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