In triangle ABC, angle ABC = 48^ , angle BCA = 71^ and AB = 9 cm. In triangle PQR, angle PQR = 48^ , angle QRP = 71^ and PQ = 9 cm. Which statement is correct?

The triangles are congruent: the third angle of each is 61^ , so a matching pair of angles sits at each end of the 9 cm side

The triangles are not congruent, because the 9 cm side is not between the two angles that were given

The triangles are congruent by SAS

The triangles are congruent: the third angle of each is 61^ , so a matching pair of angles sits at each end of the 9 cm side

The triangles are similar but not congruent

In triangle ABC and triangle PQR you are told that AB = PQ and angle ABC = angle PQR. Which extra fact proves that the two triangles are congruent, with A corresponding to P, B corresponding to Q and C corresponding to R?

The two triangles have the same area

angle BAC + angle BCA = angle QPR + angle QRP

Which one of these pairs of triangles must be congruent?

Triangle ABC with AB = 6 cm, BC = 8 cm and angle ABC = 90^ , and triangle PQR with PQ = 6 cm, QR = 8 cm and angle PQR = 90^

Two triangles that each contain a right angle and a side of 6 cm

Triangle ABC with AB = 6 cm, BC = 8 cm and angle ABC = 90^ , and triangle PQR with PQ = 6 cm, QR = 8 cm and angle PQR = 90^

Two isosceles triangles that each contain an angle of 40^

Two triangles that each have angles of 30^ , 60^ and 90^

ABCD is a quadrilateral in which AB = CD and AB is parallel to DC. The diagonals AC and BD cross at the point X. Which set of statements proves that triangle ABX is congruent to triangle CDX?

AB = CD (given); angle BAX = angle DCX (alternate angles, AB parallel to DC); angle ABX = angle CDX (alternate angles, AB parallel to DC); so ASA

AX = CX and BX = DX, because diagonals bisect each other, and AB = CD, so SSS

angle AXB = angle CXD (vertically opposite); angle BAX = angle DCX (alternate angles); angle ABX = angle CDX (alternate angles); so AAA

AB = CD, BC = AD and AC is common to both triangles, so SSS

Are congruent triangles on the Foundation paper as well as Higher?

Partly. G5, the four conditions SSS, SAS, ASA and RHS, is common content, so a Foundation candidate can be asked to pick out a congruent pair and name the condition. Pearson's content guidance for 1MA1 states that the requirement to prove two triangles are congruent is Higher tier only. A question that says 'prove' and then adds a 'hence' part is Higher tier work.

What is the difference between congruent and similar triangles?

Congruent triangles are the same size as well as the same shape, so every pair of corresponding sides is equal. Similar triangles have equal angles and corresponding sides in a fixed ratio, so one is an enlargement of the other. Congruence is the case of similarity where the scale factor is 1, which is why three matching angles prove similarity and say nothing about congruence.

Why is SSA not a congruence condition?

Two sides and an angle that is not between them can describe two different triangles. With AB = 10 cm, BC = 7 cm and angle BAC = 40 degrees, an arc of radius 7 cm drawn from B cuts the other arm of the angle at two separate points, so C has two possible positions. A right angle is the exception, and that is exactly what RHS is: the right angle, the hypotenuse and one shorter side fix the third side through Pythagoras' theorem.

How many marks is a congruence proof worth on Edexcel Higher?

Three is the usual figure for the proof: one mark for three correct statements that between them fit a condition, one for justifying those statements, and one for naming the condition. A follow-on part that uses the congruence to produce a length or an angle adds 1-2 marks. The layout is fixed, so these are among the most repeatable marks in the geometry section once you have practised writing it out.

Congruent Triangles - Edexcel GCSE Higher Maths

Typically 3-5 marks per paper, usually a 3-mark proof with a short follow-on part · Spec 1MA1

Two triangles are congruent when one could be lifted off the page and dropped exactly onto the other, so all three pairs of sides and all three pairs of angles match. Edexcel splits the work across two references on the 1MA1 specification. G5 is the list of conditions itself, SSS, SAS, ASA and RHS, and it is common content, so a Foundation candidate can be asked which of four triangles is congruent to a fifth and why. G6 is the reasoning line: apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, and use known results to obtain simple proofs. Pearson's content guidance for 1MA1 says the requirement to prove two triangles are congruent is Higher tier only, so the word 'prove' next to the word 'congruent' is a Higher tier signal.

The house question gives a figure and says 'Prove that triangle ABD is congruent to triangle ACD', for 3 marks. The mark scheme behind it wants three correct statements about matching parts, a justification for each one, and the name of the condition written down at the end, and those are separate marks: a candidate who lists three true facts and stops has one of the three. A follow-on part usually comes with it, worded 'Hence' or 'Hence, or otherwise', asking for a length or an angle that the congruence hands over, and that is 1-2 marks more. The figure is nearly always a familiar one, an equilateral or isosceles triangle cut by a perpendicular, a parallelogram or rhombus cut by a diagonal, a kite, or a circle with two tangents from a point. A calculator changes nothing here, because there is no arithmetic to speak of, so this content is as likely on Paper 1 as on Papers 2 and 3.

The reports on these questions repeat themselves. Statements come out vague, 'all the sides are the same' with no indication of which side belongs to which triangle. Justifications go missing. And the condition on the final line is wrong, most often SAS quoted where the answer is RHS, because the equal angle is not the one sitting between the pair of equal sides. Two habits fix nearly all of it. Write every statement as an equation between one named part of one triangle and one named part of the other, with the reason in brackets straight after it. Then check the angle you are relying on really does sit at the vertex where your two paired sides meet. The third habit is a negative one: before you start, note what you are being asked to prove and refuse to use it. In the equilateral triangle with a perpendicular dropped to the base, BD = DC is the consequence of the congruence, not an ingredient of it.

Worked example

ABCD is a parallelogram. The diagonals AC and BD cross at the point M. (a) Prove that triangle ABM is congruent to triangle CDM. (b) Hence show that M is the midpoint of AC.

[4 marks]

AB = CD, because opposite sides of a parallelogram are equal in length.
M1: one matching pair named part by part, with a property of the parallelogram as the reason, not 'they look equal'
AB is parallel to DC. Taking AC as a transversal, angle BAM = angle DCM (alternate angles are equal). Taking BD as a transversal, angle ABM = angle CDM (alternate angles are equal).
M1: two further statements with reasons; these are the angles at the two ends of AB and at the two ends of CD
Two angles and the side between them match, so triangle ABM is congruent to triangle CDM (ASA).
A1: the condition named, with the vertices in corresponding order, A with C, B with D and M with M
(b) Corresponding sides of congruent triangles are equal, so AM = CM. M lies on AC, so M is the midpoint of AC.
B1: the follow-on mark, earned by quoting the congruence; 'the diagonals of a parallelogram bisect each other' is the thing being proved and scores nothing

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Nothing is drawn, so each figure is set out in words: which point lies on which line, which sides are given as equal, and which vertex of one triangle corresponds to which vertex of the other. Read the correspondence before anything else, because most of the numerical questions are answered by matching a part of one triangle to its partner rather than by calculating. On the condition questions, say the three statements you would write on the paper before you choose an option.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Exam pace [2 marks]

In triangle ABC, angle ABC = $48^\circ$ , angle BCA = $71^\circ$ and AB = 9 cm. In triangle PQR, angle PQR = $48^\circ$ , angle QRP = $71^\circ$ and PQ = 9 cm. Which statement is correct?

Answer options for question 1
The triangles are not congruent, because the 9 cm side is not between the two angles that were given The triangles are congruent by SAS The triangles are congruent: the third angle of each is $61^\circ$ , so a matching pair of angles sits at each end of the 9 cm side The triangles are similar but not congruent
Question 2 Exam pace [3 marks]

A floor is covered with congruent triangular tiles. Each tile is triangle ABC, with AB = 8 cm, BC = 15 cm and CA = 17 cm. 240 tiles are used. Each tile is edged separately all the way round. Work out the total length of edging needed, in metres.

Your answer (fractions like 1/2 are fine)
Question 3 Exam pace [2 marks]

In triangle ABC and triangle PQR you are told that AB = PQ and angle ABC = angle PQR. Which extra fact proves that the two triangles are congruent, with A corresponding to P, B corresponding to Q and C corresponding to R?

Answer options for question 3
BC = QR CA = RP The two triangles have the same area angle BAC + angle BCA = angle QPR + angle QRP
Question 4 Exam pace [2 marks]

Which one of these pairs of triangles must be congruent?

Answer options for question 4
Two triangles that each contain a right angle and a side of 6 cm Triangle ABC with AB = 6 cm, BC = 8 cm and angle ABC = $90^\circ$ , and triangle PQR with PQ = 6 cm, QR = 8 cm and angle PQR = $90^\circ$ Two isosceles triangles that each contain an angle of $40^\circ$ Two triangles that each have angles of $30^\circ$ , $60^\circ$ and $90^\circ$
Question 5 Stretch [3 marks]

ABCD is a parallelogram. The point P lies on AB and the point Q lies on DC, with AP = CQ = 4 cm. Triangle APD is congruent to triangle CQB, with A corresponding to C, P corresponding to Q and D corresponding to B. QB = 9.3 cm and angle ADP = $27^\circ$ . Work out the length of PD, in cm, and the size of angle CBQ, in degrees.

PD = angle CBQ =
Question 6 Stretch [3 marks]

ABCD is a quadrilateral in which AB = CD and AB is parallel to DC. The diagonals AC and BD cross at the point X. Which set of statements proves that triangle ABX is congruent to triangle CDX?

Answer options for question 6
AB = CD (given); angle BAX = angle DCX (alternate angles, AB parallel to DC); angle ABX = angle CDX (alternate angles, AB parallel to DC); so ASA AX = CX and BX = DX, because diagonals bisect each other, and AB = CD, so SSS angle AXB = angle CXD (vertically opposite); angle BAX = angle DCX (alternate angles); angle ABX = angle CDX (alternate angles); so AAA AB = CD, BC = AD and AC is common to both triangles, so SSS

Common mistakes examiners see

Writing SAS when the equal angle is not the one between the equal sides. In the equilateral triangle ABC with D on BC and AD perpendicular to BC, the facts available are AB = AC, AD common, and a right angle at D. The right angle sits between AD and DB, not between AB and AD, so SAS is the wrong label and the condition mark goes.

Underline the two sides you have paired and check the angle sits at the vertex where those two sides meet. If it does not, look at what the angle is: a right angle with the two paired sides being the hypotenuse and one shorter side is RHS, which is the only condition that works from an angle outside the pair.
Using the conclusion as a step. Quoting BD = DC while proving that triangle ABD is congruent to triangle ACD, or quoting 'the diagonals bisect each other' inside a proof about the two triangles a diagonal makes.

Write the thing you are asked to prove at the top of the working, then rule out any statement that would follow from it. Every line you use has to come from the wording of the question or from a stated property of the named figure, such as opposite sides of a parallelogram being equal or two radii of a circle being equal.
Statements with no owner. 'All the sides are equal' or 'the angles are the same' says nothing about which part of which triangle, and Edexcel mark schemes note that this wording is ambiguous and scores zero even when the candidate knows the geometry.

Use the three-line layout every time: AB = CD (opposite sides of a parallelogram), BD is common, angle ABD = angle CDB (alternate angles, AB parallel to DC). Then the condition on a fourth line. Four lines, four seconds of thought, three marks.
Accepting AAA or SSA. Three matching angles give the right shape at any size, and two sides with a non-included angle can be satisfied twice: AB = 10 cm, BC = 7 cm and angle BAC = $40^\circ$ describes two genuinely different triangles.

Count the sides in your three statements. If there are none, you have proved similarity and nothing more, so go back and find a length. If you have two sides and the angle is opposite one of them rather than between them, the only route left is a right angle and RHS.
Reading a length off a congruence written in the wrong order. Given that triangle ABC is congruent to triangle QRP, a candidate who assumes A pairs with P writes CA = RP, when the correspondence A to Q, B to R and C to P actually gives BC = RP.

Write the two names one above the other, letter above letter, before you use the result: A over Q, B over R, C over P. A side is then read off two columns at a time, and an angle off a single column, which removes the guesswork from every 'hence' part.

Frequently asked questions

Are congruent triangles on the Foundation paper as well as Higher?: Partly. G5, the four conditions SSS, SAS, ASA and RHS, is common content, so a Foundation candidate can be asked to pick out a congruent pair and name the condition. Pearson's content guidance for 1MA1 states that the requirement to prove two triangles are congruent is Higher tier only. A question that says 'prove' and then adds a 'hence' part is Higher tier work.
What is the difference between congruent and similar triangles?: Congruent triangles are the same size as well as the same shape, so every pair of corresponding sides is equal. Similar triangles have equal angles and corresponding sides in a fixed ratio, so one is an enlargement of the other. Congruence is the case of similarity where the scale factor is 1, which is why three matching angles prove similarity and say nothing about congruence.
Why is SSA not a congruence condition?: Two sides and an angle that is not between them can describe two different triangles. With AB = 10 cm, BC = 7 cm and angle BAC = 40 degrees, an arc of radius 7 cm drawn from B cuts the other arm of the angle at two separate points, so C has two possible positions. A right angle is the exception, and that is exactly what RHS is: the right angle, the hypotenuse and one shorter side fix the third side through Pythagoras' theorem.
How many marks is a congruence proof worth on Edexcel Higher?: Three is the usual figure for the proof: one mark for three correct statements that between them fit a condition, one for justifying those statements, and one for naming the condition. A follow-on part that uses the congruence to produce a length or an angle adds 1-2 marks. The layout is fixed, so these are among the most repeatable marks in the geometry section once you have practised writing it out.

Congruent Triangles - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics