Congruence – Identical Shapes in Every Way
Congruent shapes are identical in size and shape. They are like perfect copies of each other — one could be placed exactly on top of the other (allowing flipping or rotating) and they would match perfectly.
What Does Congruent Mean?
Two shapes are congruent if they have exactly the same size and shape. Corresponding sides are equal in length and corresponding angles are equal in measure. The symbol for congruence is ≡.
Congruence and Transformations
Shapes remain congruent after these transformations (the size never changes):
| Transformation | Preserves Size? | Preserves Shape? |
|---|---|---|
| Translation (sliding) | Yes | Yes |
| Rotation (turning) | Yes | Yes |
| Reflection (flipping) | Yes | Yes |
| Enlargement (scaling) | No | Yes (similar, not congruent) |
Congruence Conditions for Triangles
Two triangles are congruent if any one of these four conditions is satisfied:
| Condition | Stands For | What It Means |
|---|---|---|
| SSS | Side-Side-Side | All three sides of one triangle equal the three sides of the other |
| SAS | Side-Angle-Side | Two sides and the included angle are equal |
| ASA | Angle-Side-Angle | Two angles and the included side are equal |
| RHS | Right angle-Hypotenuse-Side | Right-angled triangles with equal hypotenuse and one other side |
Are triangles with sides 5, 7, 9 and sides 9, 5, 7 congruent?
Both have the same three side lengths (5, 7, 9 in different order). By SSS, yes, they are congruent.
Two right triangles each have hypotenuse 13 and one leg 5. Are they congruent?
By RHS (right angle, hypotenuse = 13, side = 5), yes, they are congruent.
Common Mistakes
- AAA (three equal angles) does NOT prove congruence — it only proves similarity.
- SSA is not a valid congruence condition (two sides and a non-included angle).
- Congruent does not mean in the same orientation — a reflected shape is still congruent.
Key Takeaways
- Congruent shapes have equal size and equal shape; the congruence symbol is ≡.
- Translations, rotations, and reflections produce congruent images.
- Four conditions for triangle congruence: SSS, SAS, ASA, RHS.
- AAA proves similarity, not congruence.
Practice Questions
- State which congruence condition (SSS, SAS, ASA, or RHS) applies to each pair: (a) two triangles with sides 3, 4, 5 each; (b) two right triangles with hypotenuse 10 and one leg 6 each.
- Triangle ABC has AB = 6, BC = 8, angle B = 90°. Triangle PQR has PQ = 6, QR = 8, angle Q = 90°. Are they congruent? State the condition.
- Explain why AAA is not a congruence condition. Give an example.
- After a reflection in the y-axis, is a triangle congruent to its image? Why?
- A square has sides of 5 cm. Is every square with side 5 cm congruent to it?