Geometric proof and congruent triangles

A geometric proof is a chain of facts, each with a reason. Congruent triangles are one of the most useful tools.

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The four congruence tests

Two triangles are congruent (identical) if they match by:
SSS: three pairs of equal 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
AAA is not enough: it only shows the triangles are similar.

Writing a proof

Name the two triangles in matching order.
List three facts, each with a reason (given, common side, vertically opposite angles, alternate angles, radii of the same circle...).
State the test: 'so triangle ABM is congruent to triangle ACM (SSS)'.
Then use the congruence to prove what was asked.

Example

AB = AC and M is the midpoint of BC. Prove that angle ABM = angle ACM.
AB = AC (given)
BM = CM (M is the midpoint)
AM is common to both triangles
So triangle ABM ≡ triangle ACM (SSS), so angle ABM = angle ACM.
This proves that the base angles of an isosceles triangle are equal.
Worked example

In triangles PQR and XYZ: PQ = XY, QR = YZ and angle PQR = angle XYZ. Are they congruent?

  1. Two pairs of sides are equal.
  2. The equal angles are between those sides (the included angle).

Answer: Yes, by SAS

Key idea

SSS, SAS, ASA and RHS prove congruence; AAA does not. Every line of a proof needs a reason, and the last line states the test.

Key words
Proof
A logical argument showing a statement is always true, using algebra that works for every case.
Congruent
Exactly the same shape and size.
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Similar
The same shape but a different size: angles are equal and lengths are in the same ratio.
Vertically opposite angles
The equal angles opposite each other where two straight lines cross.
Alternate angles
Angles on opposite sides of a line that crosses two parallel lines, inside the parallel lines. They are equal (a Z shape).

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. Two triangles have two pairs of equal angles, and the sides between those angles are equal. Which test proves they are congruent?

    1. RHS
    2. ASA
    3. SSS
    4. SAS
    Show the answer

    Answer: ASA

    Angle, side between, angle: ASA.

  2. Two triangles both have angles of 30°, 60° and 90°. Must they be congruent?

    1. Yes, by RHS
    2. Yes, by AAA
    3. No, one could be an enlargement of the other
    4. Yes, because both are right-angled
    Show the answer

    Answer: No, one could be an enlargement of the other

    Equal angles give the same shape but not necessarily the same size.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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