Ever wondered why the same shape can appear in different places?
💡 In Simple Words: Two triangles are congruent when you can flip, slide or turn one so it fits exactly over the other. All their sides and angles match perfectly.
What does "congruent" actually mean?
In geometry, "congruent" is a fancy word for "exactly the same size and shape". Think of two puzzle pieces that are mirror images – if you can place one on top of the other and every edge lines up, they are congruent.
Key congruence theorems you must know
These theorems are shortcuts. They tell you which parts you need to compare so you can be sure two triangles are congruent without measuring every single side.
- SSS (Side‑Side‑Side): If all three sides of one triangle are equal to the three sides of another, the triangles are congruent.
- SAS (Side‑Angle‑Side): If two sides and the angle between them are equal in both triangles, they match.
- ASA (Angle‑Side‑Angle): If two angles and the side between them are equal, the triangles are the same.
- AAS (Angle‑Angle‑Side): If two angles and any non‑included side are equal, congruence follows.
- RHS (Right‑angle‑Hypotenuse‑Side): For right‑angled triangles, if the hypotenuse (the longest side) and one other side are equal, the triangles are congruent.
How to prove two triangles are congruent – step by step
When a question asks you to show two triangles are congruent, follow this simple flow:
Start by reading the diagram carefully. Note which sides or angles are marked equal. Then pick the theorem that fits those marks. Finally, write each statement in order: "Given…", "Hence…", and end with "∴ ΔABC ≅ ΔDEF" (the symbol ∴ means "therefore").
Worked example 1: Using SSS
Suppose you have ΔPQR and ΔXYZ. You are told that PQ = XY, QR = YZ, and PR = XZ.
- List the given equalities: PQ = XY, QR = YZ, PR = XZ.
- All three sides of one triangle match the three sides of the other – that fits the SSS theorem.
- Write the proof:
Given PQ = XY, QR = YZ, PR = XZ.
Therefore, by SSS, ΔPQR ≅ ΔXYZ.
Worked example 2: Using ASA
Imagine ΔABC and ΔDEF where ∠A = ∠D, ∠B = ∠E, and side AB = DE (the side between the two equal angles).
- Mark the equal angles and the included side.
- This matches the ASA theorem because we have two angles and the side between them.
- Proof steps:
Given ∠A = ∠D, ∠B = ∠E, AB = DE.
Hence, by ASA, ΔABC ≅ ΔDEF.
Quick comparison of the congruence theorems
| Theorem | What you need | When to use it |
|---|---|---|
| SSS | All three sides | All side lengths are given |
| SAS | Two sides + included angle | Angle is sandwiched between the two known sides |
| ASA | Two angles + included side | Side lies between the two known angles |
| AAS | Two angles + any side | Angles are known but side is not between them |
| RHS | Hypotenuse + one leg (right‑angled only) | Both triangles have a right angle |
📝 Likely Exam Questions
- State the SSS theorem and give a short proof for two triangles with sides 5 cm, 7 cm, 9 cm each.
- In ΔMNO, ∠M = 40°, ∠N = 60°, and MN = 8 cm. In ΔPQR, ∠P = 40°, ∠Q = 60°, and PQ = 8 cm. Prove the triangles are congruent.
- Explain why RHS cannot be applied to two obtuse triangles even if one side and the hypotenuse are equal.
- Given a diagram where two sides and a non‑included angle are equal, which theorem should you try first and why?
- Write a full proof using SAS for triangles where AB = CD, BC = DE, and ∠B = ∠D.