Quick revision notes · Mathematics 4024 · Chapter 31 of 35
Congruent
Congruent shapes, the four triangle conditions and using them to find lengths and angles.
3 sectionsWorked examples
Common mistakes
At a glance
Congruent shapes, the four triangle conditions and using them to find lengths and angles.
1What congruent means
- Congruent shapes are identical in shape and size. They may be rotated or reflected.
- Corresponding sides are equal and corresponding angles are equal.
- Similar shapes have equal angles but sides in proportion — different sizes are allowed.
Worked example
Two triangles have the same three angles but different sides. Are they congruent?
- Same angles only shows they are similar.
No — they are similar, not necessarily congruent
2The four conditions
- Two triangles are congruent if they match on one of these:
| Condition | What must be equal |
|---|---|
| SSS | all three sides |
| SAS | two sides and the angle BETWEEN them |
| ASA (or AAS) | two angles and a corresponding side |
| RHS | right angle, hypotenuse and one other side |
Worked example
Triangles have AB = DE, angle B = angle E and BC = EF. Are they congruent? Which condition?
- Two sides equal; the equal angle is between them.
Yes, SAS
3Using congruence
- Match the vertices in order: ABC ≅ PQR means A ↔ P, B ↔ Q, C ↔ R.
- Then the side AB = PQ, BC = QR, angle B = angle Q, and so on.
- AAA and SSA do not prove congruence.
Worked example
ABC ≅ PQR. AB = 7 cm, BC = 9 cm, angle B = 48°. Find QR and angle Q.
- BC ↔ QR.
- Angle B ↔ angle Q.
QR = 9 cm, angle Q = 48°
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Using AAA to show congruence | AAA shows similarity only |
| Matching sides by their position on the page | Match by letter order or equal angles |
| Calling any two sides and an angle SAS | The angle must be between the two sides |
★In the exam
- Name the condition (SSS, SAS, ASA, RHS) in your answer.
- Redraw the triangles the same way round if they are turned or flipped.
- Mark equal sides and angles on the diagram first.