Exactly the same
Two figures are congruent if they have the same size and the same shape. One could be placed exactly on top of the other. The symbol is ≅.
Congruent figures can be moved, turned or flipped, and they’re still congruent. These moves (translations, rotations and reflections) are called rigid transformations.
Corresponding parts
When figures are congruent, their corresponding parts match:
- Corresponding sides are equal.
- Corresponding angles are equal.
If △ABC ≅ △DEF, then the order tells you what matches:
| Matching | Equal |
|---|---|
| A ↔ D | ∠A = ∠D |
| B ↔ E | ∠B = ∠E |
| C ↔ F | ∠C = ∠F |
| AB ↔ DE | AB = DE |
| BC ↔ EF | BC = EF |
| AC ↔ DF | AC = DF |
Proving triangles congruent
You don’t need to check all six parts. These shortcuts are enough:
| Rule | You need |
|---|---|
| SSS | All three pairs of sides equal |
| SAS | Two pairs of sides equal and the angle between them |
| ASA | Two pairs of angles equal and the side between them |
| AAS | Two angles and a side not between them |
| RHS (HL) | Right angle, hypotenuse and one other side |
What doesn’t work
- AAA: three equal angles only proves the triangles are the same shape (similar), not the same size.
- SSA: two sides and an angle not between them can make two different triangles.
Congruence vs similarity
| Congruent | Similar |
|---|---|
| Same shape and same size | Same shape, any size |
| Sides equal | Sides proportional |
| Angles equal | Angles equal |
Using congruence
Once triangles are proven congruent, any corresponding parts are equal. This is a powerful tool in geometry proofs and in construction: identical roof trusses, for example, are congruent triangles.
Worked examples
Example 1: △PQR has sides 5, 7, 9. △XYZ has sides 9, 5, 7. Congruent?
All three side pairs match: yes, by SSS.
Example 2: Two triangles have sides 6 and 8 with a 40° angle between them. Congruent?
Yes, by SAS.
Example 3: Two triangles both have angles 50°, 60°, 70°. Congruent?
Not necessarily: AAA shows only similarity.
Common mistakes
- Matching vertices in the wrong order.
- Using SSA or AAA as a proof of congruence.
- Using a side that isn’t between the two angles for ASA (that’s AAS, which does work, but name it correctly).
Try it yourself
- What does ≅ mean?
- If △ABC ≅ △KLM, which side equals BC?
- Two triangles share two equal angles and the side between them. Which rule?
- Are two squares with side 4 cm congruent?
Answers: 1) is congruent to 2) LM 3) ASA 4) yes
Quick recap
- Congruent figures have the same size and shape.
- Corresponding parts of congruent figures are equal.
- Prove triangles congruent with SSS, SAS, ASA, AAS or RHS.