Congruent Triangles

What you'll learn

What it means for two triangles to be congruent.
The four GCSE congruence tests: SSS, SAS, ASA and RHS.
How to spot useful facts in diagrams involving parallelograms, circles and midpoints.
How to use congruent triangles to prove angles or sides are equal.

1. What “congruent” means

Congruent triangles are exactly the same shape and exactly the same size. They may be rotated, reflected or moved, but every matching side and every matching angle is equal.

Definition

Congruent triangles

Two triangles are congruent if all their corresponding sides and corresponding angles are equal. Corresponding means “matching in the same position”. The notation $△ABC≅△DEF\triangle ABC \cong \triangle DEF$ means A matches D, B matches E, and C matches F.

Example

Reading the matching order

You are told $△PQR≅△XZY\triangle PQR \cong \triangle XZY$ . Find the angle matching $∠QPR\angle QPR$ and the side matching $QR$ .

The letter order shows the correspondences P↔X, Q↔Z and R↔Y, so matching angles and sides can be read directly.

Match the letters in order: P matches X, Q matches Z, and R matches Y.
The angle $∠QPR\angle QPR$ has Q-P-R, so the matching angle has Z-X-Y.
Therefore $∠QPR\angle QPR$ matches $∠ZXY\angle ZXY$ , and side $QR$ matches side $Z Y$ .

2. The congruence tests

You do not need to prove all three sides and all three angles. At GCSE, you use a congruence test.

The included angle is the angle between two sides you are using. For SAS, this matters: the angle must be “sandwiched” between the two equal sides.

For SAS, the equal angle must be the included angle between the two equal sides.

Key Idea

The four tests

SSS: three corresponding sides are equal.
SAS: two corresponding sides and the included angle are equal.
ASA/AAS: two corresponding angles and one corresponding side are equal.
RHS: both triangles are right-angled, with equal hypotenuses and one other equal side.

Common Mistake

AAA is not congruence

Three equal angles only prove the triangles are the same shape, not necessarily the same size. That is similarity, not congruence.

Example

Two angles and a shared side

Triangles PQR and SQR share side $QR$ . You are given $∠QPR=70∘\angle QPR=70^\circ$ , $∠PQR=45∘\angle PQR=45^\circ$ , $∠QRS=45∘\angle QRS=45^\circ$ and $∠RQS=65∘\angle RQS=65^\circ$ . Prove the triangles are congruent.

The two triangles share side QR and have two matching angle pairs, giving ASA after the missing angles are found.

In $△PQR\triangle PQR$ , the missing angle is 65° because angles in a triangle add to 180°.
In $△SQR\triangle SQR$ , the missing angle at S is 70°.
Match the angles: 70° with 70°, 45° with 45°, and the side $QR$ is common to both triangles.
Therefore $△PQR≅△SRQ\triangle PQR \cong \triangle SRQ$ by ASA.

3. Finding equal facts in diagrams

Most proof marks come from spotting facts that are already hidden in the diagram.

Useful facts include:

A common side is equal to itself.
A parallelogram is a four-sided shape with both pairs of opposite sides parallel.
Opposite sides of a parallelogram are equal.
Diagonals of a parallelogram bisect each other, meaning they cut each other exactly in half.
Vertically opposite angles are equal when two straight lines cross.
Alternate angles are equal on parallel lines.

Example

Parallelogram split by a diagonal

KLMN is a parallelogram. The diagonal $K M$ is drawn. Prove that triangle KLM is congruent to triangle MNK.

A diagonal of a parallelogram creates two triangles with two pairs of opposite equal sides and a common side.

Opposite sides of a parallelogram are equal, so $K L = M N$ and $L M = N K$ .
The side $K M$ is common to both triangles.
The two triangles have three pairs of equal corresponding sides.
Therefore $△KLM≅△MNK\triangle KLM \cong \triangle MNK$ by SSS.

4. RHS with circles and tangents

A radius is a line from the centre of a circle to the circumference. A tangent is a line that touches a circle at exactly one point.

Key Idea

Tangent and radius fact

The radius to the point where a tangent touches a circle is perpendicular to the tangent, so it makes a 90° angle.

Example

Two tangents from one point

A circle has centre O. Points P and Q are on the circle. Lines XP and XQ are tangents. Prove that $△XPO\triangle XPO$ is congruent to $△XQO\triangle XQO$ .

Radii to tangent points make right angles, and XO is the shared hypotenuse in the two right-angled triangles.

$O P$ and $O Q$ are radii of the same circle, so $O P = O Q$ .
A radius meets a tangent at 90°, so $∠XPO=∠XQO=90∘\angle XPO=\angle XQO=90^\circ$ .
$X O$ is common to both triangles, and it is the hypotenuse in both right-angled triangles.
Therefore $△XPO≅△XQO\triangle XPO \cong \triangle XQO$ by RHS.

5. Midpoints and equal-side chains

A midpoint cuts a line segment into two equal parts. In harder proofs, you often combine midpoint facts with parallelogram facts to create equal sides.

Example

Midpoints inside a triangle

In triangle XYZ, M is the midpoint of XZ, N is the midpoint of XY, and P is the midpoint of YZ. ZMNP is a parallelogram. Prove that the triangle with vertices X, M, N is congruent to the triangle with vertices Y, N, P.

Midpoint facts and opposite sides in parallelogram ZMNP create three equal side pairs for SSS.

N is the midpoint of XY, so $X N = N Y$ .
M is the midpoint of XZ, so $X M = M Z$ . In parallelogram ZMNP, opposite sides are equal, so $M Z = N P$ . Therefore $X M = N P$ .
P is the midpoint of YZ, so $Y P = P Z$ . In parallelogram ZMNP, opposite sides are equal, so $M N = P Z$ . Therefore $M N = Y P$ .
The three pairs of matching sides are $X N = N Y$ , $X M = N P$ and $M N = Y P$ .
Therefore, in matching order, $△XMN≅△NPY\triangle XMN \cong \triangle NPY$ by SSS.

6. Using congruence to prove angles equal

Once you have proved two triangles are congruent, their matching angles and sides must be equal. This is often the final step in a proof.

Example

A kite-style quadrilateral

Quadrilateral WXYZ has $W X = W Z$ and $Y X = Y Z$ . Prove that $∠WXY=∠WZY\angle WXY=\angle WZY$ .

The kite-style quadrilateral splits into two SSS-congruent triangles sharing WY, so the angles at X and Z match.

Compare $△WXY\triangle WXY$ and $△WZY\triangle WZY$ , because they contain the two angles you want to prove equal.
You are given $W X = W Z$ and $Y X = Y Z$ .
The side $W Y$ is common to both triangles.
Therefore $△WXY≅△WZY\triangle WXY \cong \triangle WZY$ by SSS.
Corresponding angles in congruent triangles are equal, so $∠WXY=∠WZY\angle WXY=\angle WZY$ .

Tip

Perpendicular in an equilateral triangle

If a line from a vertex of an equilateral triangle meets the opposite side at 90°, compare the two smaller right-angled triangles: equal hypotenuses, common perpendicular side, then RHS.

Exam technique

In the exam

Mark equal sides and angles on the diagram as soon as you identify them.
Write the three matching facts clearly, with a reason for each one.
Finish with the exact test: “therefore the triangles are congruent by SSS/SAS/ASA/RHS”, then use corresponding angles or sides if needed.

Self review

Check yourself

Which congruence test uses the hypotenuse?
Why are three equal angles not enough to prove congruence?
In a parallelogram, which facts might help you prove two triangles congruent?

Congruent Triangles

What you'll learn

What it means for two triangles to be congruent.
The four GCSE congruence tests: SSS, SAS, ASA and RHS.
How to spot useful facts in diagrams involving parallelograms, circles and midpoints.
How to use congruent triangles to prove angles or sides are equal.

1. What “congruent” means

Congruent triangles are exactly the same shape and exactly the same size. They may be rotated, reflected or moved, but every matching side and every matching angle is equal.

Definition

Congruent triangles

Example

Reading the matching order

You are told $△PQR≅△XZY\triangle PQR \cong \triangle XZY$ . Find the angle matching $∠QPR\angle QPR$ and the side matching $QR$ .

The letter order shows the correspondences P↔X, Q↔Z and R↔Y, so matching angles and sides can be read directly.

Match the letters in order: P matches X, Q matches Z, and R matches Y.
The angle $∠QPR\angle QPR$ has Q-P-R, so the matching angle has Z-X-Y.
Therefore $∠QPR\angle QPR$ matches $∠ZXY\angle ZXY$ , and side $QR$ matches side $Z Y$ .

2. The congruence tests

You do not need to prove all three sides and all three angles. At GCSE, you use a congruence test.

The included angle is the angle between two sides you are using. For SAS, this matters: the angle must be “sandwiched” between the two equal sides.

For SAS, the equal angle must be the included angle between the two equal sides.

Key Idea

The four tests

SSS: three corresponding sides are equal.
SAS: two corresponding sides and the included angle are equal.
ASA/AAS: two corresponding angles and one corresponding side are equal.
RHS: both triangles are right-angled, with equal hypotenuses and one other equal side.

Common Mistake

AAA is not congruence

Three equal angles only prove the triangles are the same shape, not necessarily the same size. That is similarity, not congruence.

Example

Two angles and a shared side

The two triangles share side QR and have two matching angle pairs, giving ASA after the missing angles are found.

In $△PQR\triangle PQR$ , the missing angle is 65° because angles in a triangle add to 180°.
In $△SQR\triangle SQR$ , the missing angle at S is 70°.
Match the angles: 70° with 70°, 45° with 45°, and the side $QR$ is common to both triangles.
Therefore $△PQR≅△SRQ\triangle PQR \cong \triangle SRQ$ by ASA.

3. Finding equal facts in diagrams

Most proof marks come from spotting facts that are already hidden in the diagram.

Useful facts include:

A common side is equal to itself.
A parallelogram is a four-sided shape with both pairs of opposite sides parallel.
Opposite sides of a parallelogram are equal.
Diagonals of a parallelogram bisect each other, meaning they cut each other exactly in half.
Vertically opposite angles are equal when two straight lines cross.
Alternate angles are equal on parallel lines.

Example

Parallelogram split by a diagonal

KLMN is a parallelogram. The diagonal $K M$ is drawn. Prove that triangle KLM is congruent to triangle MNK.

A diagonal of a parallelogram creates two triangles with two pairs of opposite equal sides and a common side.

Opposite sides of a parallelogram are equal, so $K L = M N$ and $L M = N K$ .
The side $K M$ is common to both triangles.
The two triangles have three pairs of equal corresponding sides.
Therefore $△KLM≅△MNK\triangle KLM \cong \triangle MNK$ by SSS.

4. RHS with circles and tangents

A radius is a line from the centre of a circle to the circumference. A tangent is a line that touches a circle at exactly one point.

Key Idea

Tangent and radius fact

The radius to the point where a tangent touches a circle is perpendicular to the tangent, so it makes a 90° angle.

Example

Two tangents from one point

A circle has centre O. Points P and Q are on the circle. Lines XP and XQ are tangents. Prove that $△XPO\triangle XPO$ is congruent to $△XQO\triangle XQO$ .

Radii to tangent points make right angles, and XO is the shared hypotenuse in the two right-angled triangles.

$O P$ and $O Q$ are radii of the same circle, so $O P = O Q$ .
A radius meets a tangent at 90°, so $∠XPO=∠XQO=90∘\angle XPO=\angle XQO=90^\circ$ .
$X O$ is common to both triangles, and it is the hypotenuse in both right-angled triangles.
Therefore $△XPO≅△XQO\triangle XPO \cong \triangle XQO$ by RHS.

5. Midpoints and equal-side chains

A midpoint cuts a line segment into two equal parts. In harder proofs, you often combine midpoint facts with parallelogram facts to create equal sides.

Example

Midpoints inside a triangle

Midpoint facts and opposite sides in parallelogram ZMNP create three equal side pairs for SSS.

N is the midpoint of XY, so $X N = N Y$ .
M is the midpoint of XZ, so $X M = M Z$ . In parallelogram ZMNP, opposite sides are equal, so $M Z = N P$ . Therefore $X M = N P$ .
P is the midpoint of YZ, so $Y P = P Z$ . In parallelogram ZMNP, opposite sides are equal, so $M N = P Z$ . Therefore $M N = Y P$ .
The three pairs of matching sides are $X N = N Y$ , $X M = N P$ and $M N = Y P$ .
Therefore, in matching order, $△XMN≅△NPY\triangle XMN \cong \triangle NPY$ by SSS.

6. Using congruence to prove angles equal

Once you have proved two triangles are congruent, their matching angles and sides must be equal. This is often the final step in a proof.

Example

A kite-style quadrilateral

Quadrilateral WXYZ has $W X = W Z$ and $Y X = Y Z$ . Prove that $∠WXY=∠WZY\angle WXY=\angle WZY$ .

The kite-style quadrilateral splits into two SSS-congruent triangles sharing WY, so the angles at X and Z match.

Compare $△WXY\triangle WXY$ and $△WZY\triangle WZY$ , because they contain the two angles you want to prove equal.
You are given $W X = W Z$ and $Y X = Y Z$ .
The side $W Y$ is common to both triangles.
Therefore $△WXY≅△WZY\triangle WXY \cong \triangle WZY$ by SSS.
Corresponding angles in congruent triangles are equal, so $∠WXY=∠WZY\angle WXY=\angle WZY$ .

Tip

Perpendicular in an equilateral triangle

If a line from a vertex of an equilateral triangle meets the opposite side at 90°, compare the two smaller right-angled triangles: equal hypotenuses, common perpendicular side, then RHS.

Exam technique

In the exam

Mark equal sides and angles on the diagram as soon as you identify them.
Write the three matching facts clearly, with a reason for each one.
Finish with the exact test: “therefore the triangles are congruent by SSS/SAS/ASA/RHS”, then use corresponding angles or sides if needed.

Self review

Check yourself

Which congruence test uses the hypotenuse?
Why are three equal angles not enough to prove congruence?
In a parallelogram, which facts might help you prove two triangles congruent?

Congruent Triangles

What you'll learn

1. What “congruent” means

2. The congruence tests

3. Finding equal facts in diagrams

4. RHS with circles and tangents

5. Midpoints and equal-side chains

6. Using congruence to prove angles equal

Recap questions

You've reached the end

Congruent Triangles

What you'll learn

1. What “congruent” means

2. The congruence tests

3. Finding equal facts in diagrams

4. RHS with circles and tangents

5. Midpoints and equal-side chains

6. Using congruence to prove angles equal

Recap questions

You've reached the end

Congruent Triangles

What you'll learn

1. What “congruent” means

2. The congruence tests

3. Finding equal facts in diagrams

4. RHS with circles and tangents

5. Midpoints and equal-side chains

6. Using congruence to prove angles equal

Recap questions

You've reached the end

Congruent Triangles Revision Guide

Congruent Triangles

What you'll learn

1. What “congruent” means

2. The congruence tests

3. Finding equal facts in diagrams

4. RHS with circles and tangents

5. Midpoints and equal-side chains

6. Using congruence to prove angles equal

Recap questions

You've reached the end

Congruent Triangles Revision Guide