Congruent Triangles GCSE Maths Revision Guide

Congruent triangles GCSE Maths revision guide

In congruent triangles GCSE Maths questions, you must decide whether two triangles are exactly the same size and shape, then justify your conclusion using a valid congruence criterion. The four criteria you need are SSS, SAS, ASA and RHS.

This guide explains each criterion step by step, shows you how to write a geometrical proof and includes worked examples and exam-style questions with answers. Congruence is assessed at both Foundation and Higher tier, although Higher questions are more likely to combine it with algebra, angle facts or formal proof.

What are congruent triangles?

Two triangles are congruent if they have the same shape and the same size. Every side length and every angle in one triangle matches a corresponding side length or angle in the other.

The triangles do not need to face the same direction. One may be rotated, reflected or translated and still be congruent. Its orientation and position are irrelevant because a rigid transformation does not alter lengths or angles.

The mathematical symbol for congruence is $\cong$. For example,

$$△ABC\cong △DEF$$

means that triangle $ABC$ is congruent to triangle $DEF$. The order of the letters shows the matching vertices:

• $A$ corresponds to $D$
• $B$ corresponds to $E$
• $C$ corresponds to $F$

Consequently, $AB=DE$, $BC=EF$ and $AC=DF$. Correctly matching the vertices matters when you use congruence to find an unknown side or angle.

The four triangle congruence criteria

You do not usually need to establish all six matching measurements. Any one of the four accepted criteria provides enough information to prove that two triangles must be congruent.

SSS: side, side, side

Use SSS when all three sides of one triangle equal the three corresponding sides of another triangle.

For example, if one triangle has side lengths $5$ cm, $7$ cm and $9$ cm, and another has the same three side lengths, the triangles are congruent by SSS. The sides can appear in a different order or orientation in the diagram.

Three fixed side lengths create only one triangle shape, apart from reflection. This is why SSS is sufficient.

SAS: side, angle, side

Use SAS when two pairs of corresponding sides are equal and their included angle is equal. The included angle is the angle directly between the two known sides.

Suppose $AB=DE$, $AC=DF$ and $\angle BAC=\angle EDF$. The stated angles are between the stated sides, so

$$△ABC\cong △DEFby\; SAS.$$

Be careful with SSA, which means side, side and a non-included angle. SSA is not a general congruence criterion because the same measurements can sometimes produce two different triangles.

ASA: angle, side, angle

Use ASA when two pairs of corresponding angles and a corresponding side are equal. GCSE questions may provide the side between the angles or another corresponding side.

Once two angles are known, the third is fixed because the angles in a triangle total $180^{\circ }$. A known side then fixes the scale, so the triangle has a unique size and shape.

Some resources call the second arrangement AAS, meaning angle, angle, side. In GCSE proofs, it is normally covered by the same underlying ASA reasoning because the third angle can be calculated. Follow the terminology used by your exam board or question.

RHS: right angle, hypotenuse, side

Use RHS only when both triangles are right-angled. You must establish three facts:

1. Both triangles contain a $90^{\circ }$ angle.
2. Their hypotenuses are equal.
3. One other pair of corresponding sides is equal.

The hypotenuse is the side opposite the right angle and is always the longest side. Do not use RHS if the equal side you have called the hypotenuse is not opposite the right angle.

How to prove that triangles are congruent

A complete proof should identify the triangles, state the relevant equal facts and give the congruence criterion. Use the following process.

1. Mark the two triangles. Trace their boundaries carefully, especially in an overlapping diagram.
2. Match corresponding vertices. Look for equal sides, equal angles and shared features.
3. Write three justified facts. Each equality should come from the diagram or an established geometrical rule.
4. Check the criterion. For SAS, confirm that the angle is included. For RHS, confirm which side is the hypotenuse.
5. State the conclusion. Name the triangles in corresponding order and give SSS, SAS, ASA or RHS.

Do not rely entirely on how a printed diagram looks. GCSE diagrams are not necessarily drawn accurately, so a length or angle is equal only if it is stated, marked or follows from a valid geometrical fact.

Congruent triangles worked examples

Worked example 1: using SSS

Triangles $ABC$ and $DEF$ have the following measurements:

$$AB=8cm,BC=6cm,AC=10cm$$ $$DE=8cm,EF=6cm,DF=10cm$$

Match the sides:

$$AB=DE,BC=EF,AC=DF$$

All three corresponding side pairs are equal. Therefore,

$$△ABC\cong △DEFby\; SSS.$$

The order $ABC$ and $DEF$ is correct because $A$ matches $D$, $B$ matches $E$ and $C$ matches $F$.

Worked example 2: using SAS

In triangles $PQR$ and $XYZ$,

$$PQ=XY=7cm$$ $$PR=XZ=5cm$$ $$\angle QPR=\angle YXZ=60^{\circ }$$

The $60^{\circ }$ angle lies between the two stated sides in each triangle. Therefore,

$$△PQR\cong △XYZby\; SAS.$$

If the equal angle had not been between the two known sides, SAS would not apply.

Worked example 3: using ASA and angle facts

In triangles $ABC$ and $DCB$, $AB$ is parallel to $DC$. It is given that

$$\angle ABC=\angle DCB$$

Because $AB$ is parallel to $DC$, alternate angles give

$$\angle ACB=\angle DBC.$$

The side $BC$ is common to both triangles, so

$$BC=CB.$$

The triangles have two equal angles and a corresponding equal side. Therefore,

$$△ABC\cong △DCBby\; ASA.$$

This example shows why reasons matter. The parallel-line fact justifies one pair of equal angles, while the shared side supplies the required length.

Worked example 4: using RHS

Triangles $ABC$ and $DEF$ are right-angled at $B$ and $E$. It is given that

$$AC=DF=13cm$$

and

$$AB=DE=5cm.$$

The sides $AC$ and $DF$ are opposite the right angles, so they are the hypotenuses. Both triangles are right-angled, their hypotenuses are equal and one other corresponding side is equal. Therefore,

$$△ABC\cong △DEFby\; RHS.$$

Using congruence to find missing values

After proving two triangles congruent, you may use the fact that corresponding parts of congruent triangles are equal. This lets you equate matching sides or angles.

Suppose $△ABC\cong △DEF$ and corresponding sides have lengths

$$AB=3x+1$$

and

$$DE=16.$$

Since $AB$ corresponds to $DE$,

$$3x+1=16$$ $$3x=15$$ $$x=5.$$

The algebra is simple, but it is essential to use the order of the congruence statement to select the correct pair of sides.

Congruent triangles exam questions

Main exam-style question

Triangles $ABC$ and $ADC$ lie on opposite sides of the line $AC$. It is given that

$$AB=AD$$

and

$$\angle BAC=\angle CAD.$$

Prove that $△ABC$ and $△ADC$ are congruent. If $BC=2x+3$ and $DC=15$, find $x$.

Answer

$$AB=AD(given)$$ $$\angle BAC=\angle CAD(given)$$ $$AC=AC(common\; side)$$

The equal angle is between the two corresponding equal sides. Therefore,

$$△ABC\cong △ADCby\; SAS.$$

Corresponding sides in congruent triangles are equal, so

$$BC=DC$$ $$2x+3=15$$ $$2x=12$$ $$x=6.$$

Try 4 more congruent triangles practice questions.

Four more exam-style questions

Question 1

A triangle has side lengths $6$ cm, $8$ cm and $11$ cm. A second triangle has side lengths $11$ cm, $6$ cm and $8$ cm. Are the triangles congruent? Give a reason.

Answer: Yes. The three corresponding side lengths are equal, so the triangles are congruent by SSS.

Question 2

Triangles $PQR$ and $XYZ$ have $PQ=XY$, $PR=XZ$ and $\angle PQR=\angle XYZ$. Is this enough information to prove congruence by SAS?

Answer: No. In triangle $PQR$, the stated angle is not between $PQ$ and $PR$. The information is SSA, which is not a general congruence criterion.

Question 3

Two right-angled triangles have equal hypotenuses of $17$ cm and one corresponding pair of shorter sides equal to $8$ cm. Explain why they are congruent.

Answer: Both triangles are right-angled, their hypotenuses are equal and one other corresponding side is equal. They are congruent by RHS.

Question 4

It has been proved that $△ABC\cong △DEF$. If $AC=4y-2$ and $DF=22$, find $y$.

Answer: The order shows that $AC$ corresponds to $DF$, so

$$4y-2=22$$ $$4y=24$$ $$y=6.$$

Common mistakes in congruence questions

Using AAA as proof of congruence

Three equal angles show that triangles have the same shape, but not necessarily the same size. AAA proves similarity, not congruence. One triangle could be an enlargement of the other.

Treating SSA as SAS

For SAS, the equal angle must be between the two equal sides. If it is not, the information may allow more than one possible triangle.

Using RHS without identifying the right angle

Equal hypotenuses and another equal side are useful only when both triangles are known to be right-angled. State this explicitly in your proof.

Assuming facts from the appearance of a diagram

Lines that look equal, parallel or perpendicular are not automatically so. Use labels, matching marks and proven angle or length facts.

Omitting reasons

A list of equal sides and angles may not receive full proof marks if the reasons are missing. Useful reasons include given, common side, vertically opposite angles, alternate angles and angles in a triangle total $180^{\circ }$.

Naming vertices in the wrong order

If $A$ matches $D$, $B$ matches $F$ and $C$ matches $E$, write

$$△ABC\cong △DFE,$$

not $△DEF$. The order communicates the correspondence and affects later calculations.

Congruence compared with similarity

Congruent and similar triangles both have matching angles, but their side relationships are different.

If corresponding side lengths are $4$ cm and $8$ cm, the triangles might be similar with scale factor $2$, but they cannot be congruent. Congruent triangles always have scale factor $1$.

How to improve at congruence proofs

Practise identifying the criterion before writing a full solution. Cover the options and ask which three facts force the triangle to have one fixed size and shape.

When a diagram contains overlapping triangles, copy the triangles separately on paper and label only established information. This often makes common sides and matching vertices easier to recognise.

You should also revise the angle rules frequently used to create a proof, particularly vertically opposite angles, alternate angles, corresponding angles and the angle sum of a triangle. The GCSE Maths topics by exam board and tier guide can help you place congruence alongside the other geometry content you need to revise.

Conclusion

For congruent triangles GCSE Maths questions, learn SSS, SAS, ASA and RHS, then check that the given information satisfies every condition of the selected criterion. A strong proof states equal facts with reasons, names the triangles in corresponding order and finishes with the correct criterion.

Remember that AAA establishes similarity rather than congruence, SSA is not a general congruence test, and diagrams must not be treated as accurately drawn. Use the MathsGenie congruence lesson and practice questions to apply these checks to a range of exam-style diagrams.

Sources and referenced URLs

• https://www.mathsgenie.co.uk/congruence.html
• https://www.mathsgenie.co.uk/blog/gcse-maths-topics-by-exam-board-and-tier