GCSEMaths: FoundationEdexcelRevision NotesGeometry & MeasuresCongruence, Similarity & Geometrical ProofCongruence

Congruence (Edexcel GCSE Maths: Foundation)

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Dan

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Congruence

What is congruence?

  • Two shapes are congruent if they are identical in shape and size
    • One may be a reflection, rotation, or translation of the other
  • If one shape is an enlargement of the other, then they are not identical in size and so are not congruent

How do we prove that two shapes are congruent?

  • To show that two shapes are congruent you need to show that they are both the same shape and the same size
    • If a shape has been reflected, rotated or translated, then its image is congruent to it
  • Show that corresponding sides are the same length
  • Show that corresponding angles are the same size
  • You do not need to show that they are facing in the same direction

Examiner Tip

  • Tracing paper can help in the exam if you are unsure whether two shapes are congruent
    • Trace over one shape and then see if it fits exactly on top of the other
    • Only do this if the image is drawn to scale

Worked example

Write down the letters of the two shapes below which are congruent to A.


4-5-1-congruence-we-question

Shapes C and D are congruent to A

Congruent Triangles

What are congruent triangles?

  • Two triangles are congruent if they are the same size and shape
    • Although they may be reflections, translations or rotations of each other
  • All three angles and all three sides must be the same in both triangles

How do I prove that two triangles are congruent?

  • We only need to show that 3 of the 6 things are the same for both triangles
    • as long as they are the right three!
  • To do this we must use one of the 5 standard tests
Name Description Diagram
SAS
Side Angle Side		 Two sides and the angle between them SAS triangle
ASA
Angle Side Angle		 Two angles and the side between them ASA triangle
AAS
Angle Angle Side		 Any two angles and any side AAS triangle
SSS
Side Side Side		 All three sides SSS triangle
RHS
Right-angle Hypotenuse Side		 The hypotenuse and any other side for a right-angled triangle RHS triangle

 

Examiner Tip

  • AAA and SSA are not congruent conditions

Worked example

Prove that triangles ABC and PQR are congruent.

congruent-triangle-we

Angle ABC and angle RPQ are both 25°

Angle BAC and angle PRQ are both 90°

Line PR and line AB are both 6cm

Two angles are the same, and the lengths between them are the same

Triangles are congruent by the ASA condition

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.