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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
  • All boards
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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?

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    Bisect each other

  4. 4

    What is a proof?

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    A logical argument that shows a statement is always true

  5. 5

    Are vertically opposite angles equal?

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    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.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Prove 4 marks

    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.
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    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.

    Mark scheme

    • \(AB = DC\) 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
  2. Question 2 Prove 2 marks

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

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    Model answer

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

    Mark scheme

    • Corresponding sides of congruent triangles — M1
    • \(AM = MC\) — A1
  3. Question 3 Prove 4 marks

    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.
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    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).

    Mark scheme

    • \(AB = AC\) and \(BD = DC\) with reasons — M1
    • AD common — M1
    • SSS — A1
    • Conclusion about angles — C1
  4. Question 4 Prove 3 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • Equal sides and common side — M1
    • SSS — M1
    • Conclusion — A1
  5. Question 5 Explain 2 marks

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

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    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.

    Mark scheme

    • Not appearances or measurements — M1
    • Each step needs a mathematical reason — C1
  6. Question 6 Prove 3 marks

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

    Show answerHide answer

    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 \(AC = BD\).

    Mark scheme

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

Quick check

  1. In a proof, every statement needs a...

    1. ADiagram
    2. BMeasurement
    3. CReason
    4. DCalculator
    Show answerHide answer

    C: Reason

    A reason.

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

    1. ACorresponding angles
    2. BAlternate angles
    3. CVertically opposite angles
    4. DAngles in a triangle
    Show answerHide answer

    B: Alternate angles

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

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

    1. AThe side is common
    2. BThe side is missing
    3. CThe side is unknown
    4. DThe triangles are similar
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    A: The side is common

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

  4. Corresponding parts of congruent triangles are...

    1. AParallel
    2. BPerpendicular
    3. CDifferent
    4. DEqual
    Show answerHide answer

    D: Equal

    Equal in length or size.

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

    1. AAAA
    2. BSSS
    3. CSSA
    4. DOnly measurements
    Show answerHide answer

    B: SSS

    SSS, using the midpoint and the common side.

  6. What should the last line of a proof do?

    1. ARepeat the first line
    2. BGive a measurement
    3. CState the conclusion
    4. DAdd a new diagram
    Show answerHide answer

    C: State the conclusion

    State the conclusion that answers the question.

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