Congruent triangles GCSE Questions and Worked Answers
Congruent triangles are identical in size and shape, even when turned or reflected. Prove congruence by matching sufficient facts: SSS, SAS, ASA or RHS. State the matching sides/angles and their reasons before naming the test.
Foundation & Higher6 worked examples10 original questions
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Start with the meaning
What you need to know about congruent triangles
Cut two triangles out of paper. If you can slide, turn or flip one so that it covers the other exactly, they are congruent. That means the same size as well as the same shape. Matching corners then have equal angles, and matching edges have equal lengths. Turning a triangle does not change these facts.
See the idea first
A few measurements can guarantee an exact match
We do not always need all six measurements. Three matching side lengths already fix both the shape and the size: this test is called SSS, for side-side-side. The table lists other sufficient tests. In a written statement such as triangle ABC ≅ triangle DEF, the order pairs A with D, B with E and C with F; it tells us which measurements must match.
Which facts fix a triangle?
Test
Matching facts needed
Important condition
SSS
Three sides
Each side has its matching partner
SAS
Two sides and one angle
The angle is between those sides
ASA
Two angles and the included side
The side lies between the two angles
RHS
Right angle, hypotenuse and one other side
The hypotenuse is opposite the right angle
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Choose a congruence test
What the problem asks: Two triangles each have sides 5 cm, 6 cm and 8 cm. Are they congruent?
How to solve it: Yes, by SSS. Pair equal side lengths; a rotated or reflected drawing makes no difference.
Check the angle's position
What the problem asks: Each triangle has sides 4 cm and 7 cm with the angle between them 50°. Is this sufficient?
How to solve it: Yes, by SAS. The given angle opens between the two measured sides; specifying a different angle would not be the SAS test.
Prove triangles within a larger figure
What the problem asks: A kite ABCD has AB = AD and BC = DC. Prove triangles ABC and ADC congruent.
How to solve it: The two stated side pairs match and AC is shared by both triangles. Those three side equalities establish SSS.
A reliable routine
Write a congruence proof
Use this when asked to show two triangles match exactly or to deduce equal sides/angles. The proof must establish enough matching measurements to fix the triangle, not simply describe its appearance.
Name the two triangles in matching-vertex order.
List the equal sides and angles you know, with reasons such as given, shared side or alternate angles.
Check that these facts fit SSS, SAS, ASA or RHS, including the angle position.
Conclude congruence using that test, then identify the required matching side or angle.
Check: AAA proves similarity, not congruence. SSA does not guarantee congruence in general. RHS is a special safe case: the right angle fixes which side is the hypotenuse. If two angles and a non-included side are given, use the 180° angle sum to find the third angle and identify an ASA match.
Fully worked
Congruent triangles GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Three sides
2 marks
Question
Triangles ABC and DEF have AB = DE = 5 cm, BC = EF = 7 cm and AC = DF = 9 cm. Prove they are congruent.
The three corresponding side pairs are equal: AB = DE, BC = EF and AC = DF. Therefore triangle ABC ≅ triangle DEF by SSS. The vertex order follows those pairings.
Example 2
The included angle
2 marks
Question
AB = PQ = 6 cm, AC = PR = 8 cm, and angle BAC = angle QPR = 40°. Prove triangles ABC and PQR congruent.
The angle at A lies between AB and AC; the angle at P lies between PQ and PR. Two matching sides and their included angle agree, so triangle ABC ≅ triangle PQR by SAS.
Example 3
Two angles
3 marks
Question
Angle A = angle D = 50°, angle B = angle E = 70°, and AB = DE = 4 cm. Prove triangles ABC and DEF congruent and find angles C and F.
AB and DE lie between the two matching angles, so the triangles are congruent by ASA.
C=F=180∘−50∘−70∘=60∘
Example 4
Right-angled triangles
2 marks
Question
Two right-angled triangles each have hypotenuse 13 cm and another side 5 cm. Prove they are congruent.
The right angles match, the hypotenuses match, and a corresponding non-hypotenuse side matches. This is RHS. Specifying the hypotenuse is essential: it is the side opposite the right angle.
Example 5
A kite proof
4 marks
Question
Kite ABCD has AB = AD and BC = DC. Prove triangles ABC and ADC congruent, then deduce an equality involving angle BAC.
Given: AB = AD and BC = DC. The diagonal AC is an edge of both triangle ABC on the left and triangle ADC on the right. Matching ticks indicate equal lengths; the sketch is not a measuring diagram.
AB = AD and BC = DC are given. AC = AC is a shared side. Therefore triangle ABC ≅ triangle ADC by SSS. A matches A, B matches D, and C matches C, so angle BAC = angle DAC: the diagonal AC bisects the angle at A.
Example 6
Why AAA is insufficient
3 marks
Question
One triangle has sides 3, 4, 5 cm. Another has sides 6, 8, 10 cm. Their angles match. Are they congruent? Explain.
Each second side is twice the matching first side:
6/3=8/4=10/5=2
The triangles are similar, but not congruent because their sizes differ. Equal angles alone cannot fix the scale.
10 original questions · total 23 marks
Congruent triangles GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 28 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1
Meaning
1 mark
What two properties must congruent triangles share?
Show worked answer
The same shape and the same size. Corresponding sides and angles match exactly.
2
SSS
2 marks
Each of two triangles has sides 6, 7 and 9 cm. State a congruence test and conclusion.
Show worked answer
All three matching side lengths agree. They are congruent by SSS.
3
SAS
2 marks
Two triangles have matching sides of 5 cm and 8 cm and a matching angle of 62° between those sides. State the test.
Show worked answer
SAS: the equal angle is included between the two equal side pairs.
4
RHS
2 marks
Two right triangles have hypotenuse 10 cm and a matching shorter side 6 cm. State the test.
Show worked answer
RHS: right angle, hypotenuse and one corresponding other side agree.
5
Vertex order
2 marks
Triangle ABC ≅ triangle XYZ. Which side matches BC, and which angle matches A?
Show worked answer
A ↔ X, B ↔ Y, C ↔ Z. Therefore BC matches YZ, and angle A matches angle X.
6
Missing angle
2 marks
Triangle ABC ≅ triangle DEF. In triangle ABC, angle B = 47° and angle C = 68°. Find angle D.
Show worked answer
A=180∘−47∘−68∘=65∘
Corresponding angles agree, so D = 65°.
7
Not enough information
2 marks
Two triangles each have a 30° angle and a 5 cm side. Must they be congruent?
Show worked answer
No. One angle and one side leave freedom to change the other sides and angles. The information does not fit a sufficient congruence test.
8
Two angles and another side
3 marks
Angle A = angle D = 40°, angle B = angle E = 80°, and AC = DF = 7 cm. Prove triangles ABC and DEF congruent.
Show worked answer
C=F=180∘−40∘−80∘=60∘
Now AC and DF are the included sides between matching angles A/C and D/F. Therefore the triangles are congruent by ASA.
9
Shared side
3 marks
AB = CB and AD = CD. Prove triangles ABD and CBD congruent.
Show worked answer
AB = CB and AD = CD are given. BD = BD is shared. Hence triangle ABD ≅ triangle CBD by SSS.
10
Parallelogram proof
4 marks
ABCD is a parallelogram. Prove triangles ABC and CDA congruent using diagonal AC and alternate angles.
Given: AB is parallel to CD, and BC is parallel to AD. Use AC as the crossing line to identify the alternate angles. AC belongs to both triangles.
Show worked answer
AB is parallel to CD, so angle BAC = angle DCA. BC is parallel to AD, so angle BCA = angle DAC. AC = CA is shared and lies between those angle pairs. Therefore triangle ABC ≅ triangle CDA by ASA. Each angle equality follows from alternate angles.
Examiner-style feedback
Common congruent triangles mistakes
Trusting the sketch
Use stated or deduced equalities. Looking the same is not proof.
Using any angle for SAS
The angle must lie between the two specified sides.
Wrong correspondence
Match vertex order before deducing sides or angles.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
Same size and shape.
State matching facts and reasons.
Choose a sufficient test.
Use congruence to justify the final equality.
Quick answers
Congruent triangles FAQ
Does a reflected triangle remain congruent?
Yes. Reflection changes orientation, not lengths or angle sizes.
Why is SSA not a general test?
The side opposite a specified angle can sometimes meet the other side in two different positions, creating different triangles with the same given measurements.
G5/G6 congruence and simple proof across tiers. Do not infer a Higher-only restriction from the difficulty of a proof. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
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