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Maths · Similarity and congruence

Geometric proof and congruence

Write clear geometric proofs, using congruent triangles to prove properties of shapes such as isosceles triangles and parallelograms.

  • 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

    Name the four conditions for congruent triangles.

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    SSS, SAS, ASA, RHS

  2. 2

    What are alternate angles?

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    Equal angles between parallel lines, in a Z shape

  3. 3

    What do the diagonals of a parallelogram do?

    Show answerHide answer

    Bisect each other

  4. 4

    What is a proof?

    Show answerHide answer

    A logical argument that shows a statement is always true

  5. 5

    Are vertically opposite angles equal?

    Show answerHide answer

    Yes

Learning Objectives

  1. 1Set out a geometric proof with statements and reasons.
  2. 2Prove triangles congruent.
  3. 3Use congruent triangles to prove other facts.
  4. 4Use standard reasons for angles in parallel lines and triangles.

Writing a Proof

A proof is a chain of statements, each with a reason.

  1. 1 State what you are given

    Write down the facts from the question
  2. 2 Find equal sides and angles

    Give a reason for each: "opposite sides of a parallelogram are equal"
  3. 3 Name the congruence condition

    SSS, SAS, ASA or RHS
  4. 4 State the conclusion

    "Therefore triangle ABC is congruent to triangle DEF"
  5. 5 Use it

    Corresponding parts of congruent triangles are equal

Proving Diagonals Bisect

ABCD is a parallelogram with diagonals AC and BD meeting at M. Prove that \(AM = MC\) and \(BM = MD\).

Show the solutionHide the solution
  1. 1 \(AB = DC\) Opposite sides of a parallelogram are equal
  2. 2 Angle \(BAM\) = angle \(DCM\) Alternate angles (AB parallel to DC)
  3. 3 Angle \(ABM\) = angle \(CDM\) Alternate angles (AB parallel to DC)
  4. 4 Triangles AMB and CMD are congruent ASA (two angles and the corresponding side)
  5. 5 So \(AM = CM\) and \(BM = DM\) Corresponding sides of congruent triangles

AnswerThe diagonals of a parallelogram bisect each other, because triangles AMB and CMD are congruent (ASA).

Base Angles of an Isosceles Triangle

In triangle ABC, \(AB = AC\). D is the midpoint of BC. Prove that angle \(ABC\) = angle \(ACB\).

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  1. 1 \(AB = AC\) Given
  2. 2 \(BD = DC\) D is the midpoint of BC
  3. 3 \(AD = AD\) Common side
  4. 4 Triangles ABD and ACD are congruent SSS
  5. 5 So angle \(ABD\) = angle \(ACD\) Corresponding angles of congruent triangles

AnswerThe base angles are equal because triangles ABD and ACD are congruent (SSS).

Good Proof, Weak Proof

A good proof

  • Every statement has a reason.
  • Uses correct mathematical language.
  • Names the congruence condition.
  • Ends with a clear conclusion.

A weak proof

  • Says "it is obvious" or "it looks equal".
  • Uses measurements from a drawing.
  • Misses out a step.
  • Does not say why triangles are congruent.

Reasons You Can Use

Choose the exact words the mark scheme expects.

  • Parallel lines

    Alternate angles are equal; corresponding angles are equal; co-interior angles add to \(180^\circ\).

  • Triangles

    Angles in a triangle add to \(180^\circ\); base angles of an isosceles triangle are equal.

  • Parallelograms

    Opposite sides are equal and parallel; diagonals bisect each other.

  • Common side

    Write "AD is common" when two triangles share a side.

Complete the Proof

PQRS is a kite with \(PQ = PS\) and \(RQ = RS\). Prove that triangles PQR and PSR are congruent, and hence that angle \(PQR\) = angle \(PSR\).

1. List equal sides.

2. Name SSS.

3. Conclude.

A good answer shows: \(PQ = PS\) (given), \(RQ = RS\) (given), \(PR = PR\) (common side). The triangles are congruent by SSS. Corresponding angles are equal, so angle \(PQR\) = angle \(PSR\).

Can I...?

  1. 1Set out statements and reasons.
  2. 2Use alternate angles in a proof.
  3. 3Name a common side.
  4. 4Choose SSS, SAS, ASA or RHS.
  5. 5Use corresponding parts of congruent triangles.
  6. 6Prove diagonals bisect.
  7. 7Prove base angles are equal.
  8. 8Finish with a conclusion.

Summary & Exam Focus

  • Every step needs a reason.
  • Give the congruence condition in words or letters.
  • Corresponding parts of congruent triangles are equal.
  • End with a conclusion that answers the question.

Exam focus

ABCD is a parallelogram. The diagonals AC and BD meet at M. Prove that triangles AMB and CMD are congruent. (4 marks) (4 marks)

Write each fact with its reason: opposite sides equal, alternate angles equal. Then name the condition (ASA here).

Key terms

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

Proof
A logical argument showing a statement is always true.
Statement
A fact written in a proof.
Reason
The rule that justifies a statement.
Corresponding
In matching positions in congruent triangles.
Common side
A side shared by two triangles.
Bisect
Cut exactly in half.

Questions and answers

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

1. Exam question Prove 4 marks Easier

ABCD is a parallelogram. The diagonals AC and BD meet at M. Prove that triangles AMB and CMD are congruent.

A parallelogram ABCD with its diagonals crossing at M.

Mark scheme — 4 marks available

  • \(AB = DC\) with a reason — B1
  • One pair of equal angles with a reason — M1
  • The second pair of equal angles with a reason — M1
  • Congruent by ASA — A1

Model answer

\(AB = DC\) (opposite sides of a parallelogram). Angle \(BAM\) = angle \(DCM\) (alternate angles, AB parallel to DC). Angle \(ABM\) = angle \(CDM\) (alternate angles). So the triangles are congruent by ASA.

2. Exam question Prove 2 marks Easier

Using the congruent triangles from the last question, prove that \(AM = MC\).

Mark scheme — 2 marks available

  • Corresponding sides of congruent triangles — M1
  • \(AM = MC\) — A1

Model answer

Triangles AMB and CMD are congruent, so corresponding sides are equal. AM corresponds to CM, so \(AM = MC\).

3. Exam question Prove 4 marks Easier

Triangle ABC is isosceles with \(AB = AC\). D is the midpoint of BC. Prove that angle \(ABC\) = angle \(ACB\).

An isosceles triangle ABC with AB equal to AC and D the midpoint of BC, with AD drawn.

Mark scheme — 4 marks available

  • \(AB = AC\) and \(BD = DC\) with reasons — M1
  • AD common — M1
  • SSS — A1
  • Conclusion about angles — C1

Model answer

\(AB = AC\) (given). \(BD = DC\) (D is the midpoint). \(AD\) is common. So triangles ABD and ACD are congruent by SSS, and angle \(ABC\) = angle \(ACB\) (corresponding angles).

4. Exam question Prove 3 marks Easier

PQRS is a kite with \(PQ = PS\) and \(RQ = RS\). Prove that angle \(PQR\) = angle \(PSR\).

Mark scheme — 3 marks available

  • Equal sides and common side — M1
  • SSS — M1
  • Conclusion — A1

Model answer

\(PQ = PS\) and \(RQ = RS\) (given). \(PR\) is common. The triangles PQR and PSR are congruent by SSS, so angle \(PQR\) = angle \(PSR\).

5. Exam question Explain 2 marks Easier

Explain why "the triangles look the same" is not a good reason in a proof.

Mark scheme — 2 marks available

  • Not appearances or measurements — M1
  • Each step needs a mathematical reason — C1

Model answer

A proof must use exact facts, not appearances or measurements. Each step needs a mathematical reason such as SSS or a parallel-line rule.

6. Exam question Prove 3 marks Easier

ABCD is a rectangle. The diagonals AC and BD are drawn. Prove that AC = BD.

Mark scheme — 3 marks available

  • Equal sides and right angles — M1
  • SAS — M1
  • \(AC = BD\) — A1

Model answer

\(AB = DC\) (opposite sides of a rectangle). \(BC\) is common to triangles ABC and DCB. Angle \(ABC\) = angle \(DCB = 90^\circ\). So triangles ABC and DCB are congruent by SAS, and \(AC = BD\).

7. Multiple choice 1 mark Easier

In a proof, every statement needs a...

  1. A Diagram
  2. B Measurement
  3. C Reason Correct
  4. D Calculator

Why: A reason.

8. Multiple choice 1 mark Core

Which reason justifies angle BAM = angle DCM in a parallelogram?

  1. A Corresponding angles
  2. B Alternate angles Correct
  3. C Vertically opposite angles
  4. D Angles in a triangle

Why: AB is parallel to DC, so they are alternate angles.

9. Multiple choice 1 mark Core

When two triangles share a side, you write...

  1. A The side is common Correct
  2. B The side is missing
  3. C The side is unknown
  4. D The triangles are similar

Why: The shared side is common, so it is equal in both.

10. Multiple choice 1 mark Core

Corresponding parts of congruent triangles are...

  1. A Parallel
  2. B Perpendicular
  3. C Different
  4. D Equal Correct

Why: Equal in length or size.

11. Multiple choice 1 mark Core

To prove the base angles of an isosceles triangle are equal, you can show two triangles are congruent using...

  1. A AAA
  2. B SSS Correct
  3. C SSA
  4. D Only measurements

Why: SSS, using the midpoint and the common side.

12. Multiple choice 1 mark Stretch

What should the last line of a proof do?

  1. A Repeat the first line
  2. B Give a measurement
  3. C State the conclusion Correct
  4. D Add a new diagram

Why: State the conclusion that answers the question.