Geometry and measures

Congruent triangles and geometric proof

7 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher2 of 7 from Foundation papers

Revision guide by The Full Marks

Congruent triangles and geometric proof: what you need to know

Congruent shapes are exactly the same shape and size. You need to spot congruent triangles, say which condition proves it, and use known facts to write short geometric proofs. Proofs are mostly Higher, but spotting congruence appears on Foundation too.

Both tiers need the conditions and simple reasoning, but full written proofs are mostly asked on Higher papers.

Key facts

SSS: all three sides equal.
SAS: two sides and the angle between them equal.
ASA (or AAS): two angles and a corresponding side equal (the side must be in the same position in both triangles).
RHS: right angle, hypotenuse and one other side equal.
AAA does not prove congruence (the triangles could be different sizes), and nor does two sides with an angle that is not between them.

Writing a congruence proof

Name the two triangles with matching vertices in the same order.
State three pairs of equal sides or angles, with a reason for each (given, common side, alternate angles, and so on).
Finish with the condition: 'so the triangles are congruent (SAS)'.

Other short proofs (angles, isosceles triangles, quadrilaterals)

Use letters, not measured or made-up numbers: a proof must work for every case.
Write one fact per line, each with its reason (e.g. 'angles on a straight line add to $180^\circ$ ', 'alternate angles are equal').
End by stating exactly what you were asked to prove.

Worked examples

Example 1 Foundation · grade 4

Triangle $ABC$ has $AB = 7$ cm, $BC = 9$ cm and angle $B = 40^\circ$ . Triangle $PQR$ has $PQ = 7$ cm, $QR = 9$ cm and angle $Q = 40^\circ$ . Are they congruent?

Show the solution

Two pairs of sides are equal: $AB = PQ$ and $BC = QR$ .
The $40^\circ$ angle is between those sides in both triangles.
So yes, they are congruent (SAS).

Example 2 Higher · grade 7

Triangle $ABC$ is isosceles with $AB = AC$ . $M$ is the midpoint of $BC$ . Prove that angle $ABC =$ angle $ACB$ .

Show the solution

Compare triangles $ABM$ and $ACM$ .
$AB = AC$ (given).
$BM = CM$ ( $M$ is the midpoint of $BC$ ).
$AM$ is common to both triangles.
So triangles $ABM$ and $ACM$ are congruent (SSS).
So matching angles in the congruent triangles are equal: angle $ABM =$ angle $ACM$ , i.e. angle $ABC =$ angle $ACB$ .

Common mistakes

Using AAA as a reason. Equal angles only prove the triangles are similar.
Using SAS when the angle is not between the two sides.
Stating equal sides without a reason, or assuming the thing you are asked to prove.
When picking congruent triangles from a set, choosing two that share the same numbers without checking the angle is in the same position.

Exam tips

A full proof needs three correct pairs, each with a reason, plus the condition. Marks are lost for a missing reason even if the idea is right.
If the question then asks you to explain something else, use 'corresponding sides/angles of congruent triangles are equal'.

Questions people ask

What are the four conditions for congruent triangles?

SSS, SAS, ASA (or AAS) and RHS. Any one of them is enough to prove two triangles are congruent.

What is the difference between congruent and similar?

Congruent shapes are identical in shape and size. Similar shapes have the same shape, but one is an enlargement of the other.

Exam questions: Congruent triangles and geometric proof (7, easiest first)

Congruent triangles and geometric proof2 marksGrade 1CalculatorFoundation

November 2018 · Paper 3 Foundation · Q9

Name one shape on a grid and pick out the two congruent shapes.

Show answer

(a)	Trapezium	1 mark
(b)	C and D	1 mark

Worked solution Checked against the mark scheme

(a) Shape E has exactly one pair of parallel sides, so it is a trapezium.

(b) Congruent shapes are exactly the same shape and size (they may be turned). C and D match.

Watch out: Choosing two shapes that are similar (same shape, different size) rather than congruent.

Congruent triangles and geometric proof1 markGrade 4No calculatorFoundation

November 2020 · Paper 1 Foundation · Q23

Pick out the two congruent triangles from four using their sides and angles.

Show answer

Answer	A and D	1 mark

Worked solution Checked against the mark scheme

Triangle D has angles 80° and 45°, so its third angle is $180 - 80 - 45 = 55°$ .

In D the 10 cm side lies between the 45° and 55° angles, just as in A. Same two angles and the side between them (ASA), so A and D are congruent.

Watch out: Choosing B and C because both show 8 cm and 10 cm.

Congruent triangles and geometric proof1 markGrade 5No calculatorHigher

November 2020 · Paper 1 Higher · Q4

Pick out the two congruent triangles from four labelled triangles.

Show answer

Answer	A and D	1 mark

Worked solution Checked against the mark scheme

Triangle A: the 10 cm side lies between angles of 55° and 45°.

Triangle D: the third angle is $180 - 80 - 45 = 55°$ , and its 10 cm side joins the 45° corner to that 55° corner.

So both have a 10 cm side between angles of 55° and 45° (ASA), and A and D are congruent.

Watch out: Choosing B and C because they both have 8 cm and 10 cm sides, without checking the angle is in the same position.

Congruent triangles and geometric proof3 marksGrade 7CalculatorHigher

November 2022 · Paper 3 Higher · Q20

Prove two overlapping triangles inside an isosceles triangle are congruent.

Show answer

Answer	Congruent by SAS: $AE = DE$ , angle $EAC$ = angle $EDB$ , $AC = DB$	3 marks

Worked solution Checked against the mark scheme

$AE = DE$ (given).

Triangle $ADE$ is isosceles, so angle $EAC$ = angle $EDB$ (base angles).

Using the ratio, $AC = 1 + 2 = 3$ parts and $DB = 1 + 2 = 3$ parts, so $AC = DB$ .

Two sides and the included angle match, so the triangles are congruent (SAS).

Watch out: Using AAA, which does not prove congruence.

Congruent triangles and geometric proof3 marksGrade 8CalculatorHigher

November 2024 · Paper 2 Higher · Q18

Prove that the line from the centre of a circle to a point equidistant from two others on the circle bisects the angle there.

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Answer	Triangles ABO and CBO are congruent (SSS), so angle ABO = angle CBO	3 marks

Worked solution Checked against the mark scheme

$BA = BC$ (given).

$OA = OC$ (radii of the circle).

$OB$ is common to both triangles.

So triangles $ABO$ and $CBO$ are congruent (SSS), and angle $ABO$ = angle $CBO$ : $OB$ bisects angle $ABC$ .

Watch out: Stating sides are equal without giving reasons, or assuming what must be proved.

Congruent triangles and geometric proof5 marksGrade 8CalculatorHigher

June 2018 · Paper 3 Higher · Q21

Prove two triangles inside a parallelogram are congruent, then use this to explain why two lines are parallel.

Show answer

(a)	$AD = BC$ (opposite sides of a parallelogram), angle $DAP$ = angle $BCQ$ (opposite angles of a parallelogram), angle $ADP$ = angle $CBQ = 90°$ , so the triangles are congruent (ASA)	3 marks
(b)	$AP = QC$ from the congruence and $AP$ is parallel to $QC$ , so $APCQ$ is a parallelogram and its opposite sides $AQ$ and $PC$ are parallel	2 marks

Worked solution Checked against the mark scheme

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

Angle $DAP$ = angle $BCQ$ : opposite angles of a parallelogram are equal.

Angle $ADP$ = angle $CBQ = 90°$ (given).

So triangles $ADP$ and $CBQ$ are congruent (ASA).

(b) From (a), $AP = CQ$ . $AP$ lies on $AB$ and $CQ$ lies on $DC$ , which are parallel. A quadrilateral with one pair of sides equal and parallel is a parallelogram, so $APCQ$ is a parallelogram and $AQ$ is parallel to $PC$ .

Watch out: Stating the triangles are congruent without giving a reason for each pair, or quoting SAS/RHS wrongly.

Congruent triangles and geometric proof3 marksGrade 8No calculatorHigher

Specimen · Paper 1 Higher · Q17

Prove two triangles inside an isosceles triangle are congruent.

Show answer

Answer	Congruent by SAS: $QR$ is common, angle $PQR$ = angle $PRQ$ , and $QS = RT$	3 marks

Worked solution Checked against the mark scheme

$PQ = PR$ , so triangle $PQR$ is isosceles and angle $SQR$ = angle $TRQ$ (base angles).

$S$ and $T$ are midpoints of equal sides, so $QS = \frac{1}{2}PQ = \frac{1}{2}PR = RT$ .

$QR$ is a side of both triangles.

So triangles $QTR$ and $RSQ$ have two sides and the included angle equal: congruent by SAS.

Watch out: Stating the sides and angle are equal without giving reasons.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

November 2024 · Paper 1 Foundation · Q21: Find an expression for an angle in a parallelogram using parallel-line angle facts, with reasons.
June 2024 · Paper 2 Higher · Q22: Prove that two tangents drawn to a circle from the same external point are equal in length.
June 2023 · Paper 3 Foundation · Q22: Prove a triangle is isosceles using parallel-line and triangle angle facts with reasons.
June 2022 · Paper 3 Higher · Q15: Prove two triangles formed by intersecting chords in a circle are similar, giving reasons.
November 2019 · Paper 1 Higher · Q22: Prove two triangles inside a cyclic quadrilateral are congruent, using circle theorems and an equilateral triangle.
November 2019 · Paper 1 Foundation · Q29: Use similar triangles to find a missing side, then use a pair of congruent triangles to find a length.
Specimen · Paper 2 Higher · Q14: Use Pythagoras and similar right-angled triangles to find the length of the hypotenuse of the larger triangle.