125. SSS Congruence Criterion · Three equal sides guarantee congruence

Try to break it

Drag P anywhere on the canvas. The blue triangle slides and turns, but its three side lengths never change. Try to stretch one side to a new length — you can't. With all three sides matching the red triangle, SSS locks the shape.

How you build it

Construct a triangle from three given sides (SSS).

• Draw segment AB — the base of your triangle, of length c.
• With centre A and radius b, draw an arc above AB. Every point on this arc is exactly distance b from A — so C must lie on it.
• With centre B and radius a, draw a second arc. Where the two arcs cross is point C — the only point at distance b from A AND distance a from B.
• Draw segment AC by joining A to the point where the two arcs intersect. Its length is exactly b.
• Draw segment BC by joining B to the same intersection point. Length BC = a. Triangle ABC is now complete — and because we used exactly the three given side lengths, any triangle built this way is congruent to ABC (SSS).

The proof, step by step

Prove that three equal sides make the triangles congruent (SSS).

• Given: In ΔABC and ΔPQR, AB = PQ, BC = QR, and AC = PR.
• Place ΔPQR on ΔABC such that vertex P coincides with A and side PQ lies along AB.
• Since PQ = AB, vertex Q must coincide with B.
• Since AC = PR and BC = QR, vertex R must fall exactly on C.
• Therefore, ΔABC ≅ ΔPQR by the SSS Congruence Test.

Worked example

In ΔABC and ΔPQR, AB = 5 cm, BC = 6 cm, AC = 7 cm and PQ = 5 cm, QR = 6 cm, PR = 7 cm. Are the triangles congruent? If yes, state the congruence criterion.

All three corresponding sides are equal (AB=PQ, BC=QR, AC=PR). By the SSS Congruence Test, ΔABC ≅ ΔPQR.

• Yes, by SAS
• Yes, by SSS — correct
• No, they are not congruent
• Yes, by ASA