Congruent Triangles –

Congruent triangles are triangles that are exactly the same shape and size. Proving that two triangles are congruent is a key GCSE Maths skill, tested on AQA, Edexcel, and OCR. You must know the four conditions and be able to write a structured proof using correct mathematical language. This guide covers each condition, walks through proof-style worked examples, and flags the mistakes that lose marks.

What Are Congruent Triangles?

Two triangles are congruent if one can be placed exactly on top of the other — they have the same side lengths and the same angles. To prove congruence, you do not need to show all six measurements match. Instead, you show that one of four sufficient conditions is satisfied.

Key Formulas

The included angle in SAS sits between the two known sides. The included side in ASA sits between the two known angles.

Step-by-Step Method

1. Identify the two triangles you need to prove congruent.
2. List what you know about each triangle — sides, angles, shared edges, or given information.
3. Match the information to one of the four conditions: SSS, SAS, ASA, or RHS.
4. Write the proof in a structured way, stating each pair of equal elements with a reason.
5. Conclude by stating: "Therefore triangle ABC is congruent to triangle DEF by [condition]."

Worked Example 1 — Foundation Level

Question: Triangle PQR has sides PQ = 5 cm, QR = 7 cm, and PR = 9 cm. Triangle XYZ has sides XY = 5 cm, YZ = 7 cm, and XZ = 9 cm. Are the triangles congruent? Give a reason.

Working:

PQ = XY = 5 cm

QR = YZ = 7 cm

PR = XZ = 9 cm

All three pairs of corresponding sides are equal.

Answer: Yes, triangle PQR is congruent to triangle XYZ by SSS (three sides equal).

Worked Example 2 — Higher Level

Question: In the diagram, ABCD is a parallelogram. Prove that triangle ABD is congruent to triangle CDB.

Working:

AB = CD (opposite sides of a parallelogram are equal)

AD = CB (opposite sides of a parallelogram are equal)

BD = BD (common side)

Therefore triangle ABD is congruent to triangle CDB by SSS.

Answer: Triangles ABD and CDB are congruent by SSS.

Worked Example 3 — Exam Style

Question: In triangle ABC, M is the midpoint of BC. AM is perpendicular to BC. Prove that triangle ABM is congruent to triangle ACM.

Working:

BM = CM (M is the midpoint of BC)

Angle AMB = angle AMC = 90° (AM is perpendicular to BC)

AM = AM (common side)

Therefore triangle ABM is congruent to triangle ACM by SAS (two sides and the included angle are equal).

Answer: Triangle ABM is congruent to triangle ACM by SAS.

Common Mistakes

• Using SSA or AAA. Two sides and a non-included angle (SSA) is not a valid congruence condition. Three equal angles (AAA) proves similarity, not congruence. Only SSS, SAS, ASA, and RHS are accepted.
• Not identifying the included angle or side. For SAS, the angle must be between the two sides. For ASA, the side must be between the two angles. Getting this wrong means your proof is invalid.
• Forgetting to state reasons. Each line of a congruence proof must have a reason, such as "opposite sides of a parallelogram" or "common side". A bare statement without justification will not earn full marks.

Exam Tips

• Start your proof by clearly naming the two triangles you are comparing.
• Always write your proof as a list of paired equalities, each with a reason.
• End with a clear conclusion: "Therefore triangle ... is congruent to triangle ... by [SSS/SAS/ASA/RHS]."
• If the question involves a right angle and you know the hypotenuse plus one other side, use RHS — it is often overlooked but is perfectly valid.
• Look for shared sides (common edges) — these are free equalities you can use.

Practice Questions

Q1 (Foundation): Triangle DEF has DE = 4 cm, angle DEF = 60°, and EF = 6 cm. Triangle GHI has GH = 4 cm, angle GHI = 60°, and HI = 6 cm. Are they congruent? State the condition.

Q2 (Foundation): Triangle JKL has angle J = 50°, JK = 8 cm, and angle K = 70°. Triangle MNO has angle M = 50°, MN = 8 cm, and angle N = 70°. Are they congruent?

Q3 (Higher): ABCD is a rectangle. Prove that triangle ABC is congruent to triangle CDA.

Practise congruent triangle proofs with instant feedback free on GCSEMathsAI.

Related Topics

• Similar Shapes and Scale Factors
• Congruence and Similarity
• Pythagoras' Theorem

Summary

• Two triangles are congruent if they satisfy one of four conditions: SSS, SAS, ASA, or RHS. A congruence proof must list matching pairs of sides or angles with reasons and end with a clear conclusion naming the condition used. Remember that SSA and AAA are not valid congruence conditions. Look for common sides, parallel-line properties, and given right angles to build your proof.