Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Congruence and geometric proof. 5 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Congruent shapes are exactly the same shape and size. You need the four conditions for congruent triangles (SSS, SAS, ASA and RHS), and you use them, with angle facts and the properties of shapes, to write geometric proofs.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
3
Identify congruent shapesCongruent shapes are identical in shape and size, even if one is turned over or rotated.
5
Know the four congruence conditionsSSS, SAS, ASA (or AAS) and RHS; three equal angles is not enough.
5
Choose the condition for two trianglesMatch the equal sides and angles, then name the condition that applies.
Notes
Congruent shapes
Congruent shapes are identical: same shape and same size, so all corresponding sides and angles are equal.
One shape may be a rotation, reflection or translation of the other.
Similar shapes have equal angles but can be different sizes. Congruent shapes are similar with scale factor 1.
Conditions for congruent triangles
SSS: all three pairs of sides are equal.
SAS: two pairs of sides and the angle between them (the included angle) are equal.
ASA: two pairs of angles and a pair of corresponding sides are equal (the side may be between the angles or not, sometimes written AAS).
RHS: both have a right angle, equal hypotenuses and one other pair of equal sides.
Three equal angles (AAA) only shows the triangles are similar. Two sides and an angle that is not between them do not prove congruence.
Writing a congruence proof
Name the triangles with corresponding vertices in the same order, e.g. triangle ABC and triangle CDA.
Write three pairs of equal sides or angles, one per line, each with a reason: given, common side, radii are equal, vertically opposite angles are equal, alternate angles are equal, opposite sides of a parallelogram are equal.
Finish with the conclusion and the condition, e.g. 'so triangle ABC is congruent to triangle CDA (SSS)'.
Once triangles are congruent, all their corresponding sides and angles are equal, which you can use to prove further results.
Geometric proof
Start from facts you know, give a reason for every step and end with exactly what you were asked to show.
Use letters (e.g. let angle BAC = x) to prove a result in general; a numerical example is not a proof.
Cheatsheet
Congruent = same shape and same size
SSS: three pairs of equal sides
SAS: two sides and the included angle
ASA (or AAS): two angles and a corresponding side
RHS: right angle, hypotenuse and one other side
AAA shows similar only, not congruent
Common reasons: given; common side; radii are equal; vertically opposite angles are equal; alternate angles are equal
How to answer each type of question
Decide whether triangles are congruent and give the condition
2 marks5
Match up the equal sides and angles in the two triangles.
Check whether an angle is between the two sides you are using.
Name the condition (SSS, SAS, ASA or RHS).
Example. Triangle PQR has PQ = 7 cm, angle PQR = 48° and QR = 10 cm. Triangle XYZ has XY = 10 cm, angle XYZ = 48° and YZ = 7 cm. Are the two triangles congruent? Give a reason for your answer.
Show the model answer
Yes (B1) Both have sides of 7 cm and 10 cm with the 48° angle between them, so they are congruent by SAS (B1)
Shortcuts and memory tricks
There are only four tests: SSS, SAS, ASA, RHS. If yours is not one of these, it is not a valid reason.
For SAS, point along side, angle, side in order: the angle must be sandwiched between the two sides.
A side shared by both triangles is a free pair: write 'common side'.
Writing the vertices in matching order makes it easy to read off corresponding sides.
Where marks are lost
Using AAA as a reason for congruence (it only shows similarity).
Using SAS when the angle is not between the two sides.
Stating facts without reasons: 'AB = CD' on its own does not score.
Forgetting to write the condition (SSS, SAS, ASA or RHS) at the end.
Using the result you are trying to prove as one of your facts.
Exam technique
Set out a proof as three lines of 'fact because reason', then a conclusion with the condition; this earns the communication mark.
Use only information that is given or that you have proved, not what the diagram looks like.
If you are asked to prove two lengths or angles are equal, look for congruent triangles that contain them.
Quick recall
Cover the answers and test yourself. The app has these as flashcards that come back just before you'd forget them.
Write down the condition for congruence when two sides and the angle between them are equal.
SAS
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
(a) Ahmed says, 'If two triangles have the same three angles, they must be congruent.' Explain why Ahmed is wrong.[1]
(b) Write down the condition for congruence when two sides and the angle between them are equal.[1]
Show the answer and mark scheme
(a)Answer: Triangles with the same angles can be different sizes (they are similar, not necessarily congruent), e.g. equilateral triangles with sides 2 cm and 5 cm.
C1 for explaining that the triangles could be different sizes (an enlargement), e.g. two equilateral triangles with different side lengths
Worked solution: An equilateral triangle of side 2 cm and one of side 5 cm both have angles 60°, 60°, 60°, but they are not the same size, so they are not congruent.
(b)Answer: SAS
B1 for SAS
Worked solution: SAS (side, angle, side).
Question 2Medium7 marks
The diagram shows a circle, centre O. AB is a chord of the circle. M is the midpoint of AB. Diagram NOT accurately drawn
[object Object]
(a) Prove that triangle OAM is congruent to triangle OBM.[3]
(b) Hence prove that OM is perpendicular to AB.[2]
(c) OA = 9 cm and AB = 14 cm. Work out the length of OM. Give your answer as a surd in its simplest form.[2]
Show the answer and mark scheme
(a)Answer: OA = OB (radii), AM = BM (M is the midpoint of AB), OM is common, so the triangles are congruent (SSS).
C1 for OA = OB, with the reason that they are radii (of the same circle)
C1 for AM = BM, with the reason that M is the midpoint of AB
C1 for OM common to both triangles, and the conclusion that the triangles are congruent (SSS)
Worked solution: In triangles OAM and OBM: OA = OB (radii of the same circle) AM = BM (M is the midpoint of AB) OM = OM (common side) So triangle OAM is congruent to triangle OBM (SSS).
(b)Answer: Angle OMA = angle OMB (corresponding angles of congruent triangles). AMB is a straight line, so angle OMA + angle OMB = 180°, so each angle is 90°.
C1 for angle OMA = angle OMB, with the reason that they are corresponding angles of congruent triangles
C1 for angle OMA + angle OMB = 180° (angles on a straight line), so angle OMA = 90° and OM is perpendicular to AB
Worked solution: From part (a), angle OMA = angle OMB (corresponding angles of congruent triangles). AMB is a straight line, so angle OMA + angle OMB = 180° (angles on a straight line add up to 180°). So 2 × angle OMA = 180° and angle OMA = 90°. Therefore OM is perpendicular to AB.
(c)Answer: \(OM = 4\sqrt{2}\) cm
M1 for AM = 7 and \(OM^2 = 9^2 - 7^2\) (= 32)
A1 for \(4\sqrt{2}\)
Worked solution: AM = ½ × 14 = 7 cm. Since OM is perpendicular to AB, triangle OAM is right-angled at M: \(OM = \sqrt{9^2 - 7^2} = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\) cm