Chapter 4 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.4 Chapter 4

Congruent triangles

Two triangles have two equal sides and one equal angle. Must they be exactly the same?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Show two triangles are congruent using SSS, SAS, ASA or RHS
  • Use congruent triangles to prove a simple fact

Two shapes are congruent when one fits exactly on the other. You may need to slide, turn or flip it first. Congruent shapes have the same size and the same form. Their corresponding sides and angles are equal. Those are the parts that match each other.

Two triangles drawn to the same size. Triangle ABC has sides marked with one, two and three ticks. Triangle DEF is turned and flipped but has the same ticks on the matching sides. Corners are named A, B, C, D, E and F.ABCDEF
Matching ticks mean equal sides. Matching arcs mean equal angles.

To show two triangles are congruent you do not need all six facts. Three facts, in the right pattern, are enough. There are four patterns. Each has a short name.

SSS means three sides. If all three sides of one triangle equal the three sides of another, the triangles are congruent. A triangle with sides of 4 cm, 5 cm and 6 cm can only be drawn one way.

Two triangles with sides of 4 cm, 5 cm and 6 cm, marked with one, two and three ticks. The second triangle is turned. Corners are named P, Q, R, X, Y and Z.4 cm5 cm6 cmPQR4 cm5 cm6 cmXYZ
SSS: three pairs of equal sides.

SAS means two sides and the angle between them. The angle must sit between the two sides. Then the triangle is fixed, because the two sides and the angle between them decide where the third side goes.

Two triangles. Each has a side of 5 cm with one tick, a side of 8 cm with two ticks, and a 60° angle between them marked with an arc. The second triangle is turned and flipped. Corners are named S, T, U, V, W and X.60°5 cm8 cmSTU60°5 cm8 cmVWX
SAS: two pairs of equal sides, and the equal angles are between them.

ASA means two angles and the side between them. If two angles are equal, the third angle is equal too, because the three angles of a triangle add up to 180°. So two pairs of equal angles and one pair of equal matching sides is enough, wherever that side is. This version is called AAS.

Two triangles. Each has a side of 7 cm with one tick at the bottom. The angle at the left end of that side has one arc and the angle at the right end has two arcs, in both triangles. The second triangle is turned and flipped. Corners are named J, K, L, M, N and P.7 cmJKL7 cmMNP
ASA: two equal angles and the side between them.

RHS is for right angled triangles only. It means a right angle, the hypotenuse and one other side. The hypotenuse is the longest side, opposite the right angle. A right angle, the hypotenuse and one more side fix the triangle.

Two right angled triangles. Each has a right angle mark, a hypotenuse of 5 cm with one tick, and a side of 3 cm with two ticks. The second triangle is turned. Corners are named D, E, F, G, H and J.3 cm5 cmDEF3 cm5 cmGHJ
RHS: a right angle, the hypotenuse and one other side.

One pattern does not always work. Two sides and an angle that is not between them is called SSA. Look at what can happen.

Two triangles that share a side of 5 cm with one tick, a side of 8 cm with two ticks, and a 30° angle marked with an arc at the corner A. The first triangle has a long third side. The second has a short third side. They are not the same size. Corners are named A, B, C, D, E and F.30°5 cm8 cmACB30°5 cm8 cmDFE
SSA: the same two sides and the same angle, but two different triangles.
Two triangles. Triangle ABC has a side AB of 5 cm with one tick, a side BC of 7 cm with two ticks, and an angle of 40° at B. Triangle DEF has a side DE of 5 cm with one tick, a side EF of 7 cm with two ticks, and an angle of 40° at E. The corners are named A, C, B, D, F and E.40°7 cm5 cmACB40°7 cm5 cmDFE
Triangle ABC and triangle DEF.

Worked example

Triangle ABC has AB = 5 cm, BC = 7 cm and angle B = 40°. Triangle DEF has DE = 5 cm, EF = 7 cm and angle E = 40°. Are the triangles congruent?

  1. Match the sides. AB matches DE and BC matches EF. Both pairs are equal.
  2. Angle B is between AB and BC. Angle E is between DE and EF. Both angles are 40°.
  3. Two sides and the angle between them are equal. So the triangles are congruent by SAS.

Answer: Yes, by SAS

Congruent triangles are useful because they prove things. A proof is a chain of reasons that shows a fact must be true. Take an isosceles triangle, which has two equal sides. Draw a line from the top corner to the middle of the base. It cuts the triangle into two triangles.

Two triangles drawn side by side, the two halves of an isosceles triangle. Triangle BMA has a base BM of 6 cm marked with two ticks, a side AB of 10 cm marked with three ticks, and a side AM marked with one tick. Triangle CMA is the mirror image with the same ticks. The corners are named B, M, A, C, M and A.10 cm6 cmBMA10 cm6 cmCMA
The two halves have three pairs of equal sides.

Worked example

In triangle ABC, AB = AC = 10 cm and BC = 12 cm. M is the middle of BC. Show that triangles ABM and ACM are congruent, and find BM.

  1. AB = AC = 10 cm. The problem says so.
  2. BM = MC, because M is the middle of BC. Each is half of 12 cm, which is 6 cm.
  3. AM is a side of both triangles, so it is equal to itself.
  4. Three pairs of sides are equal, so the triangles are congruent by SSS. Also BM is 6 cm.

Answer: SSS, BM is 6 cm

Because the triangles are congruent, their matching angles are equal. So angle ABM equals angle ACM. This proves that the angles at the base of any isosceles triangle are equal.

See it three ways

With things

Cut a triangle out of paper. Trace round it on a second sheet and cut out the trace. Lay one on top of the other. They match exactly.

As a picture

Two right angled triangles with sides of 3 cm, 4 cm and 5 cm. Each has a right angle mark. The second is turned. Corners are named K, L, M, N, P and Q.3 cm4 cm5 cmKLM3 cm4 cm5 cmNPQ

In symbols

Congruent triangles have equal matching sides and equal matching angles. For example KL = NP, LM = PQ and MK = QN.

Try it

Build a triangle from three strips

You need: three strips of paper or card, 3 cm, 4 cm and 5 cm long, a friend, a pencil.

  1. Lay the 3 cm and 4 cm strips so that their ends touch at a corner.
  2. Swing the 5 cm strip across to join the other two ends.
  3. Ask your friend to build a triangle from three strips of the same lengths. Lay one triangle on the other.

What you will see: The two triangles match exactly, however you place the strips. Three sides fix the triangle. That is SSS.

If you cannot try it, this is what happens.
Two triangles made from strips of 3 cm, 4 cm and 5 cm. Each has a right angle mark. One is turned. Corners are named S, T, U, V, W and X.3 cm4 cm5 cmSTU3 cm4 cm5 cmVWX

A wrong idea many people have

Two sides and an angle are always enough to show triangles are congruent.

Why it feels right: Two sides and an angle are three facts, and SAS uses exactly that.

The test that breaks it: Look at the picture of SSA. Both triangles have a side of 5 cm, a side of 8 cm and a 30° angle. But the third sides are different, so the triangles are not the same.

What is true: The angle must be between the two sides for SAS. If it is not between them, two different triangles can sometimes fit the facts, so SSA proves nothing. RHS works because the angle is 90° and the side opposite it is the longest.

Key words

congruent
having exactly the same size and form. One shape fits exactly on the other.
corresponding
matching. Corresponding sides and angles sit in the same place in each shape.
SSS
three sides of one triangle equal three sides of another.
SAS
two sides and the angle between them are equal.
ASA
two angles and the side between them are equal. If the side is not between them, the test is called AAS and it works too.
RHS
right angle, hypotenuse and one other side are equal.
proof
a chain of reasons showing a fact must be true.

Back to your guess

The answer is C: Not always. When the angle is between the two sides they must be. When it is not, they may differ.

It depends where the angle is. When the equal angle is between the two equal sides, the triangles are the same. When it is not, two different triangles can sometimes be made, so this pattern, SSA, is not a test. RHS is the one exception you will use: a right angle with the hypotenuse fixes the triangle. This chapter shows which sets of three facts are enough.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The marks show that three pairs of sides are equal. Which test shows that the triangles are congruent?

    Two triangles with sides of 4 cm, 5 cm and 6 cm, marked with one, two and three ticks on matching sides. The corners are named A, B, C, D, E and F.4 cm5 cm6 cmABC4 cm5 cm6 cmDEF
    Tap your answer.

    1. Not this one. SAS needs an angle between two of the sides. No angle is marked, only the three sides. Go to M4.4.4. Try another.

    2. Yes, that is right. All three pairs of sides are equal, and that is the SSS test.

    3. Not this one. RHS is for right angled triangles. No right angle is marked here. Go to M4.4.4. Try another.

    See the answer

    SSS

    All three pairs of sides are equal, and that is the SSS test.

    • If you chose A: SAS needs an angle between two of the sides. No angle is marked, only the three sides. Go to M4.4.4.
    • If you chose C: RHS is for right angled triangles. No right angle is marked here. Go to M4.4.4.

  2. These two triangles have two equal sides and an equal angle. Are they congruent?

    Two triangles. Each has a side of 5 cm marked with one tick, a side of 7 cm marked with two ticks, and a 40° angle at corner A. The first triangle has a longer third side. The corners are named A, C, B, D, F and E.40°5 cm7 cmACB40°5 cm7 cmDFE
    Tap your answer.

    1. Not this one. The angle of 40° is not between the 5 cm and 7 cm sides. SAS needs the angle between the two sides. Go to M4.4.4. Try another.

    2. Not this one. SSS needs all three sides equal. The third sides are different lengths in the picture. Go to M4.4.4. Try another.

    3. Not this one. Triangles that look alike can differ in size. A reason must be one of the four tests. Go to M4.4.4. Try another.

    4. Yes, that is right. The equal angle is not between the two equal sides. That is SSA, which is not a test of congruence. The picture shows two different triangles with different third sides.

    See the answer

    No. Their third sides are different lengths. SSA cannot prove two triangles congruent.

    The equal angle is not between the two equal sides. That is SSA, which is not a test of congruence. The picture shows two different triangles with different third sides.

    • If you chose A: The angle of 40° is not between the 5 cm and 7 cm sides. SAS needs the angle between the two sides. Go to M4.4.4.
    • If you chose B: SSS needs all three sides equal. The third sides are different lengths in the picture. Go to M4.4.4.
    • If you chose C: Triangles that look alike can differ in size. A reason must be one of the four tests. Go to M4.4.4.

  3. Each triangle has a right angle. The marks show equal hypotenuses and one other equal side. Which test shows that the triangles are congruent?

    Two right angled triangles. Each has a right angle mark, a hypotenuse of 5 cm marked with one tick, and a side of 3 cm marked with two ticks. The corners are named A, B, C, D, E and F.3 cm5 cmABC3 cm5 cmDEF
    Tap your answer.

    1. Not this one. ASA needs two equal angles and a side. Here one angle is marked, and there are two sides. Go to M4.4.4. Try another.

    2. Not this one. SAS needs the angle between the two equal sides. The right angle is not between the hypotenuse and the 3 cm side. Go to M4.4.4. Try another.

    3. Yes, that is right. The triangles each have a right angle, equal hypotenuses and one more equal side. That is RHS.

    See the answer

    RHS

    The triangles each have a right angle, equal hypotenuses and one more equal side. That is RHS.

    • If you chose A: ASA needs two equal angles and a side. Here one angle is marked, and there are two sides. Go to M4.4.4.
    • If you chose B: SAS needs the angle between the two equal sides. The right angle is not between the hypotenuse and the 3 cm side. Go to M4.4.4.

  4. The marks show two equal angles and the side between them. Which test shows that the triangles are congruent?

    Two triangles. Each has a side of 7 cm with one tick. The angle at one end of that side has one arc and the angle at the other end has two arcs. The corners are named A, B, C, D, E and F.7 cmABC7 cmDEF
    Tap your answer.

    1. Yes, that is right. Two angles are equal and the side between them is equal. That is ASA.

    2. Not this one. SSS needs three equal sides. Only one side is marked here, with two angles. Go to M4.4.4. Try another.

    3. Not this one. RHS needs a right angle. None is marked here. Go to M4.4.4. Try another.

    See the answer

    ASA

    Two angles are equal and the side between them is equal. That is ASA.

    • If you chose B: SSS needs three equal sides. Only one side is marked here, with two angles. Go to M4.4.4.
    • If you chose C: RHS needs a right angle. None is marked here. Go to M4.4.4.

  5. Triangles ABM and ACM are congruent. AB = AC = 13 cm and BC = 10 cm. M is the middle of BC. How long is BM?

    Tap your answer.

    1. Not this one. That is the length of AB. BM is part of BC, the base of the triangle. Go to M4.4.4. Try another.

    2. Not this one. That is the whole base BC. M is the middle of BC, so BM is only half of it. Go to M4.4.4. Try another.

    3. Yes, that is right. M is the middle of BC, so BM is half of BC. Half of 10 cm is 5 cm.

    See the answer

    5 cm

    M is the middle of BC, so BM is half of BC. Half of 10 cm is 5 cm.

    • If you chose A: That is the length of AB. BM is part of BC, the base of the triangle. Go to M4.4.4.
    • If you chose B: That is the whole base BC. M is the middle of BC, so BM is only half of it. Go to M4.4.4.

  6. From an earlier chapter

    A side of 6 cm is enlarged by scale factor 4. How long is the image side, in centimetres?

    See the answer

    24

    An enlargement multiplies each length by the scale factor: 6 × 4 = 24 cm.

  7. Triangle ABC has sides of 5 cm, 6 cm and 9 cm. Triangle DEF is congruent to it and has sides of 6 cm, 9 cm and x cm. What is x?

    See the answer

    5

    Congruent triangles have the same three side lengths. DEF has 6 cm and 9 cm, so the third side matches the 5 cm of ABC.

  8. Triangle ABC has AB = AC and angle B = 70°. Triangles ABM and ACM are congruent, so angle C is also 70°. What is angle A, in degrees?

    See the answer

    40

    The angles of a triangle add up to 180°. Angle A = 180° − 70° − 70° = 40°.

  9. Triangle ABC has angles of 50° and 60°. Triangle DEF also has angles of 50° and 60°, and one pair of matching sides is equal. What is the third angle of each triangle, in degrees?

    See the answer

    70

    The angles of a triangle add up to 180°. The third angle is 180° − 50° − 60° = 70° in both triangles. So all three angles match, and with the equal side the triangles are congruent by AAS.

Keep it

Copy this into your exercise book.

Congruent triangles match exactly. Tests: SSS, SAS (angle between), ASA, RHS. SSA is not a test, but RHS works. Use them to prove things, such as equal base angles.

Draw: Two matching triangles with ticks on the equal sides and arcs on the equal angles.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.