Teach Me Maths

Congruence

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What congruent shapes are, how to pick them out from a set, the four conditions for congruent triangles, and how to find a missing side or angle by matching the corresponding parts, including triangles that share a side.

Congruence in the exam

Two shapes are congruent when they are exactly the same shape and exactly the same size. One of them may be turned round or flipped over, but every length and every angle matches.

In the exam you may be shown a set of shapes and asked to write down the letters of the two that are congruent, for 1 mark. You may also be told that two triangles are congruent and asked to work out a missing length or angle, for 1 or 2 marks.

Some questions ask which facts are enough to show that two triangles are congruent. There are four conditions to learn: SSS, SAS, ASA and RHS.

The diagrams are usually not drawn accurately, so don't measure them. Use the lengths and angles you are given.

Same shape, same size

Turning a shape (a rotation), flipping it over (a reflection) or sliding it to a new place (a translation) does not change any of its lengths or angles. So the new shape is congruent to the old one.

Making a shape bigger or smaller (an enlargement) keeps the angles the same but changes the lengths. The new shape is similar, not congruent.

Congruent shapes always have the same area. But two shapes with the same area are not always congruent, so check the shape as well.

Worked example 1: Picking out congruent shapes

Here are five shapes on a square grid. Which of the shapes B, C, D and E are congruent to shape A?

  1. Shape A is a column of 3 squares with 1 more square at the bottom on the right.
  2. Shape B is shape A turned a quarter turn. Shape C is shape A flipped over. Both are the same shape and size as A, so both are congruent to A.

  3. Your turn. How many squares make shape D?

    Shape D is made of 4 squares, the same number as shape A, but they make a T shape. It is a different shape, so it is not congruent to A.

  4. Shape E is shape A with every length doubled. It is the same shape but a different size, so it is not congruent to A.
  5. Answer: shapes B and C.

Matching sides and angles

In two congruent shapes, each side has a corresponding side in the other shape: the side that matches it. Corresponding sides are equal, and corresponding angles are equal.

When a question says that triangle ABC is congruent to triangle PQR, the letters match in order: A with P, B with Q and C with R. So side AB matches side PQ, side BC matches side QR, and angle B matches angle Q.

Don't match sides by where they sit on the page. The second triangle may be turned or flipped, so its bottom side might not match the bottom side of the first triangle. Use the letters, the marks on the sides or the lengths and angles you know.

Worked example 2: Using the order of the letters

Triangle JKL is congruent to triangle RST. J matches R, K matches S and L matches T. Work out the values of x and y.

  1. Write the matching letters in pairs: J and R, K and S, L and T.
  2. Side ST joins S and T. These match K and L, so ST matches side KL.
  3. So x=8, because KL is 8 cm.

  4. Your turn. Side RT joins R and T, which match J and L. What is the value of y?

    RT matches JL, so y=7

  5. Answer: x=8 and y=7

The four conditions for congruent triangles

To show that two triangles are congruent, you don't need to check all three sides and all three angles. Any one of these four conditions is enough.

  • SSS (side, side, side): all three sides are equal.
  • SAS (side, angle, side): two sides and the angle between them are equal.
  • ASA (angle, side, angle): two angles and the side between them are equal.
  • RHS (right angle, hypotenuse, side): both triangles have a right angle, their hypotenuses are equal and one other side is equal.

The hypotenuse is the longest side of a right-angled triangle. It is the side opposite the right angle.

If two angles of one triangle are equal to two angles of the other, the third angles are equal too, because angles in a triangle add up to 180^\circ. So for ASA, work out the third angle first when you need it.

What is not enough

Three equal angles are not enough. An enlargement keeps all the angles but changes the lengths. So two triangles with angles of 50^\circ, 60^\circ and 70^\circ can be different sizes.

For SAS, the angle must be between the two sides. Sides of 7 cm and 10 cm with a 35^\circ angle between them make just one triangle. If the 35^\circ angle is somewhere else, two such triangles do not have to be congruent.

For ASA, the side must be in the same place. In both triangles, the equal side must sit between the same two angles.

Worked example 3: Which triangle is congruent

Here are three triangles, P, Q and R. One of the triangles Q and R is congruent to triangle P. Which one?

  1. In triangle P, the 9 cm side is between angles of 40^\circ and 65^\circ
  2. In triangle Q, the 9 cm side is between angles of 40^\circ and 75^\circ. Its angles are the same as P's, but its 9 cm side is in a different place, so Q is a different size. It is not congruent to P.
  3. In triangle R, the 75^\circ angle is opposite the 9 cm side. The angle at the other end of the 9 cm side is not marked.

  4. Your turn. Angles in a triangle add up to 180^\circ. Work out the angle at the other end of the 9 cm side in triangle R.

    180-40-75=65^\circ

  5. So in triangle R, the 9 cm side is between angles of 40^\circ and 65^\circ, just as in triangle P. The triangles are congruent by ASA.
  6. Answer: triangle R.

Marks on the sides

Sides marked with the same number of dashes are equal. A side with one dash matches the other side with one dash, a side with two dashes matches the other side with two dashes, and so on.

To match an angle, look at the two sides it sits between. The angle between the one-dash and two-dash sides in one triangle matches the angle between the one-dash and two-dash sides in the other.

Worked example 4: Finding an angle from the marks

The two triangles are congruent. Sides with the same number of dashes are equal. Work out the size of the angle marked y.

  1. Angle y is between the side with one dash and the side with three dashes.
  2. In the first triangle, the 58^\circ angle is between the one-dash and two-dash sides, and the 74^\circ angle is between the two-dash and three-dash sides.
  3. So the angle that matches y, between the one-dash and three-dash sides, is the third angle of the first triangle. It is not marked.

  4. Your turn. Angles in a triangle add up to 180^\circ. Work out the third angle of the first triangle.

    180-58-74=48^\circ

  5. Answer: y=48^\circ

Congruent triangles inside a shape

In harder questions, two congruent triangles are joined together, so they share a side or make one bigger shape. Mark every length you know on the diagram, then use the matching sides to fill in the rest.

Watch for a side that is split into two parts. If the two parts match each other, each part is half of the whole side. If you know the whole side and one part, subtract to find the other part.

When two triangles share a side, that side matches itself. Check which way round it goes. If B matches D and D matches B, the shared side BD still matches itself, but its two ends swap over.

Worked example 5: A triangle split in two

D is a point on BC. Triangle ABD is congruent to triangle ACD, with A matching A, B matching C and D matching D. AC =13 cm, AD =12 cm and BC =10 cm. Work out the perimeter of triangle ABD.

  1. The perimeter of triangle ABD is AB + BD + AD, and AD is 12 cm.
  2. AB joins A and B, which match A and C. So AB matches AC, and AB is 13 cm.

  3. Your turn. BD matches CD, so they are equal. Together they make BC, which is 10 cm. Work out the length of BD.

    BD is 10\div2=5 cm.

  4. The perimeter of triangle ABD is 13+5+12=30 cm.
  5. Answer: 30 cm

Worked example 6: Two triangles sharing a side

The line BD splits quadrilateral ABCD into two triangles. Triangle ABD is congruent to triangle CDB, with A matching C, B matching D and D matching B. AD =5 cm, angle ABD =40^\circ and angle ADB =80^\circ. Work out the length of BC and the size of angle ABC.

  1. Write the matching letters in pairs: A and C, B and D, D and B. The shared side BD matches itself, with its ends swapped.
  2. Side BC joins B and C. These match D and A, so BC matches side DA. So BC =5 cm
  3. Angle ABC is made of two parts: angle ABD, which is 40^\circ, and angle DBC.
  4. Angle DBC has B in the middle, between D and C. These match D in the middle, between B and A. So angle DBC matches angle BDA, which is 80^\circ.

  5. Your turn. Add the two parts. Work out the size of angle ABC.

    Angle ABC =40^\circ+80^\circ=120^\circ

  6. Answer: BC =5 cm and angle ABC =120^\circ

Common mistakes

  • Picking a triangle that only looks the same. Check every length and angle, and where each one is, before you pair two triangles.
  • Matching sides by their place on the page. The second shape may be turned or flipped, so match sides using the letters, the marks or the angles.
  • Saying a turned or flipped shape is not congruent. Rotations, reflections and translations keep the same shape and size.
  • Calling an enlargement congruent. A bigger or smaller copy is similar, not congruent.
  • Using three equal angles as the reason. Equal angles do not fix the size, so you also need a matching side.
  • Using an angle that is not between the two sides. For SAS, the angle must sit between the two equal sides.

Practice questions

More on this topic: Congruence worksheet with full solutions