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Maths · Similarity and congruence

Congruence

Recognise congruent shapes, use the four conditions for congruent triangles, and use congruence to find missing lengths and angles.

  • 6 key terms
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Teacher resources

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Student handouts

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Warm-up

Answer each one, then check.

  1. 1

    What do the angles in a triangle add up to?

    Show answerHide answer

    \(180^\circ\)

  2. 2

    What does isosceles mean?

    Show answerHide answer

    Two equal sides

  3. 3

    What is a right-angled triangle's longest side called?

    Show answerHide answer

    The hypotenuse

  4. 4

    Do reflections keep the size of a shape?

    Show answerHide answer

    Yes

  5. 5

    What do the angles on a straight line add up to?

    Show answerHide answer

    \(180^\circ\)

Learning Objectives

  1. 1Explain what congruent means.
  2. 2Use SSS, SAS, ASA and RHS to show triangles are congruent.
  3. 3Explain why AAA and SSA do not prove congruence.
  4. 4Use congruence to find missing sides and angles.

CONGRUENT

Two shapes are congruent if they are exactly the same shape and size.

One can be moved onto the other by a rotation, a reflection, a translation, or a combination.

The Four Conditions

Learn the letters and what they mean.

  • SSS

    Means: All three sides equal. Example: 5, 6, 7 and 5, 6, 7

  • SAS

    Means: Two sides and the included angle equal. Example: 5, \(40^\circ\), 7 and 5, \(40^\circ\), 7

  • ASA

    Means: Two angles and a corresponding side equal. Example: \(50^\circ\), 6, \(70^\circ\)

  • RHS

    Means: Right angle, hypotenuse and one other side equal. Example: Right angle, 10, 6

Proving Two Triangles Congruent

Triangle ABC has \(AB = 5\) cm, \(BC = 7\) cm and angle \(ABC = 50^\circ\). Triangle PQR has \(PQ = 5\) cm, \(QR = 7\) cm and angle \(PQR = 50^\circ\). Are the triangles congruent?

Show the solutionHide the solution
  1. 1 Two pairs of equal sides \(AB = PQ\) and \(BC = QR\)
  2. 2 The angle between them is equal too Angle \(B\) = angle \(Q = 50^\circ\)
  3. 3 Two sides and the included angle SAS

AnswerYes, the triangles are congruent by SAS.

Conditions That Do NOT Work

  • AAA

    Three equal angles give similar triangles, but they can be different sizes.

  • SSA

    Two sides and a non-included angle can give two different triangles.

  • Only sides

    Two sides alone are not enough.

  • Angles not between the sides

    If the angle is not between the two sides, it is not SAS.

Using Congruence to Find a Length

Triangles ABC and DEF are congruent by SSS, with A matching D, B matching E and C matching F. \(AB = 6\) cm, \(BC = 8\) cm, \(DF = 10\) cm. Find \(AC\) and \(DE\).

Show the solutionHide the solution
  1. 1 Corresponding sides are equal \(AB = DE\), \(BC = EF\), \(AC = DF\)
  2. 2 Find AC \(AC = DF = 10\) cm
  3. 3 Find DE \(DE = AB = 6\) cm

Answer\(AC = 10\) cm and \(DE = 6\) cm.

Writing the Reason

A congruence answer needs three things.

  • The equal parts

    Write which sides or angles are equal, with the reason for each.

  • The condition

    State SSS, SAS, ASA or RHS.

  • The conclusion

    "So triangle ABC is congruent to triangle DEF."

Congruent or Not?

Decide which pairs of triangles are congruent and name the condition. (a) 4, 5, 6 and 6, 4, 5 (b) sides 5 and 8 with an angle of \(30^\circ\) between them, and sides 5 and 8 with an angle of \(40^\circ\) between them (c) angles \(40^\circ\), \(60^\circ\), \(80^\circ\) in both triangles, but one has a side of 3 cm and the other a side of 6 cm.

1. Match the equal parts.

2. Name the condition.

A good answer shows: (a) Congruent by SSS. (b) Not congruent: the included angles differ. (c) Not congruent: AAA does not prove congruence; the sides are different sizes, so the triangles are similar.

Can I...?

  1. 1Say what congruent means.
  2. 2Recall SSS, SAS, ASA and RHS.
  3. 3Decide whether two triangles are congruent.
  4. 4Explain why AAA is not enough.
  5. 5Explain why SSA is not enough.
  6. 6Find corresponding sides and angles.
  7. 7Use congruence to find a length.
  8. 8Write a clear reason.

Summary & Exam Focus

  • Congruent means identical in shape and size.
  • Conditions: SSS, SAS, ASA (or AAS), RHS.
  • AAA and SSA do not prove congruence.
  • Always give the condition as your reason.

Exam focus

Triangle ABC has \(AB = 6\) cm, \(AC = 8\) cm and angle \(A = 50^\circ\). Triangle PQR has \(PQ = 6\) cm, \(PR = 8\) cm and angle \(P = 50^\circ\). Show that the two triangles are congruent. (3 marks) (3 marks)

Name the pairs of equal parts and check the angle is between the two sides. Then state SAS.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Congruent
Exactly the same shape and size.
Corresponding
In matching positions in two shapes.
Included angle
The angle between two given sides.
Hypotenuse
The longest side of a right-angled triangle.
SSS, SAS, ASA, RHS
The four conditions for congruent triangles.
Similar
The same shape but a different size.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Give a reason 3 marks Easier

The diagram shows three triangles A, B and C. Two of the triangles are congruent. Write down which two, and give a reason.

Three triangles: A and B each have sides 6 cm and 8 cm with a 50 degree angle between them, and C has sides 6 cm and 8 cm with a 40 degree angle between them.

Mark scheme — 3 marks available

  • A and B — B1
  • Two sides and the included angle equal — M1
  • SAS — A1

Model answer

Triangles A and B are congruent by SAS: two sides (6 cm and 8 cm) and the included angle \(50^\circ\) are equal.

2. Exam question Show that 3 marks Easier

Triangle ABC has \(AB = 5\) cm, \(BC = 7\) cm and angle \(ABC = 50^\circ\). Triangle PQR has \(PQ = 5\) cm, \(QR = 7\) cm and angle \(PQR = 50^\circ\). Show that triangle ABC is congruent to triangle PQR.

Mark scheme — 3 marks available

  • Equal sides stated — M1
  • Equal included angle stated — M1
  • SAS and conclusion — A1

Model answer

\(AB = PQ = 5\) cm, \(BC = QR = 7\) cm, and angle \(ABC\) = angle \(PQR = 50^\circ\). Two sides and the included angle are equal, so the triangles are congruent by SAS.

3. Exam question Give a reason 2 marks Easier

Two triangles both have angles of \(40^\circ\), \(60^\circ\) and \(80^\circ\). Ben says they must be congruent. Explain why Ben is not necessarily correct.

Mark scheme — 2 marks available

  • AAA is not a congruence condition — M1
  • The triangles could be different sizes — C1

Model answer

Equal angles (AAA) do not prove congruence. The triangles could be different sizes, so they are similar but not necessarily congruent.

4. Exam question Work out 2 marks Easier

Triangles ABC and DEF are congruent, with A matching D, B matching E and C matching F. \(AB = 6\) cm, \(BC = 8\) cm and \(DF = 10\) cm. Write down the length of \(AC\).

Mark scheme — 2 marks available

  • Corresponding sides identified — M1
  • 10 — A1

Model answer

\(AC = DF = 10\) cm, because AC and DF are corresponding sides.

5. Exam question Explain 3 marks Easier

Triangle ABC is right-angled at B, with hypotenuse \(AC = 10\) cm and \(AB = 6\) cm. Triangle PQR is right-angled at Q, with hypotenuse \(PR = 10\) cm and \(PQ = 6\) cm. Explain why the triangles are congruent.

Mark scheme — 3 marks available

  • Right angle in each — B1
  • Hypotenuse and one other side equal — M1
  • RHS — A1

Model answer

Each triangle has a right angle, a hypotenuse of 10 cm and another side of 6 cm. This is RHS, so the triangles are congruent.

6. Exam question Decide 2 marks Easier

Triangle ABC has sides 4 cm, 5 cm and 6 cm. Triangle DEF has sides 6 cm, 4 cm and 5 cm. Are the triangles congruent? Give a reason.

Mark scheme — 2 marks available

  • Yes — B1
  • SSS — B1

Model answer

Yes, all three sides are equal (SSS).

7. Multiple choice 1 mark Easier

What does congruent mean?

  1. A The same shape only
  2. B The same shape and size Correct
  3. C The same size only
  4. D The same angles only

Why: The same shape and the same size.

8. Multiple choice 1 mark Core

Which is a condition for congruent triangles?

  1. A AAA
  2. B SSA
  3. C SAS Correct
  4. D ASS

Why: SAS, with the angle between the two sides.

9. Multiple choice 1 mark Core

Two triangles have the same three angles. Are they definitely congruent?

  1. A Yes, always
  2. B Yes, if right-angled
  3. C Yes, if isosceles
  4. D No Correct

Why: No: they could be different sizes, so they may only be similar.

10. Multiple choice 1 mark Core

In the condition RHS, R stands for...

  1. A Right angle Correct
  2. B Ratio
  3. C Rotation
  4. D Reflection

Why: A right angle.

11. Multiple choice 1 mark Core

Triangles are congruent by SSS. If \(AB = DE = 5\), what else do you know?

  1. A Nothing
  2. B The other sides and all angles match Correct
  3. C They are different sizes
  4. D They have different angles

Why: The other two pairs of corresponding sides are equal, and so are all the angles.

12. Multiple choice 1 mark Stretch

For SAS, the angle must be...

  1. A Opposite the longest side
  2. B Any angle
  3. C Between the two sides Correct
  4. D The smallest angle

Why: Between the two given sides.