Congruency: Congruent Triangles
123. The Coincidence Test · When two triangles land perfectly on top of each other
DEF coincides with ABC when D=A, E=B, and F=C.
Two triangles are congruent when all three sides AND all three angles of one match the corresponding sides and angles of the other. Congruent triangles can be exactly superimposed — same shape, same size.
What this lesson covers
Try to break it
Drag D, E, and F away from A, B, C. The triangles stop covering each other. Congruence isn't just about shape — the vertices have to match in correspondence. Only when D↔A, E↔B, F↔C do the triangles coincide.
How you build it
Construct any triangle PQR on the canvas.
- Place point P, the first vertex of your triangle.
- Place point Q, the second vertex. Keep it a clear distance from P.
- Place point R, the third vertex. Make sure R is not on the line PQ — otherwise the three points won't form a triangle.
- Draw segment PQ, the first side of the triangle.
- Draw segment QR, the second side of the triangle.
- Draw segment RP to complete triangle PQR.
The proof, step by step
Prove that triangle DEF coincides exactly with triangle ABC.
- Two triangles are congruent if they coincide completely when placed one over the other.
- This means all corresponding sides and angles are equal in length and measure.
- The symbol for congruency is ≅ or ≡.
Worked example
If ΔABC ≅ ΔDEF, and AB = 5 cm, BC = 6 cm, and AC = 7 cm, what is the length of EF?
Since ΔABC ≅ ΔDEF, corresponding sides are equal. BC corresponds to EF, so EF = BC = 6 cm.
- 5 cm
- 6 cm — correct
- 7 cm
- Cannot be determined