Sheet № 155 · Foundation + Higher · AQA · Edexcel · OCR
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 proo
§Key definitions
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.
Answer:
Yes, triangle PQR is congruent to triangle XYZ by SSS (three sides equal).
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.
§Formulas to memorise
SSS — three sides of one triangle equal three sides of the other
SAS — two sides and the included angle are equal
ASA — two angles and the included side are equal
RHS — right angle, hypotenuse, and one other side are equal
Identify the two triangles — you need to prove congruent.
List what you know — about each triangle — sides, angles, shared edges, or given information.
Match the information — to one of the four conditions: SSS, SAS, ASA, or RHS.
Write the proof — in a structured way, stating each pair of equal elements with a reason.
Conclude — by stating: "Therefore triangle ABC is congruent to triangle DEF by [condition]."
Worked example
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:
⚠ 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.