GCSEMathsAQARevision Notes4. Geometry & MeasuresCongruence, Similarity & Geometrical ProofCongruence

Congruence (AQA GCSE Maths)

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Congruence

What does congruent mean?

  • In maths, 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
  • You do not need to show that they are facing in the same direction
  • If a shape has been reflected, rotated or translated, then its image is congruent to it

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 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

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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 we prove that two triangles are congruent?

  • Happily we don’t have to show that all 3 sides and all 3 angles are the same in each triangle
  • We only need to show that 3 of the 6 things are the same
    • as long as they are the right three!
  • To do this we must use one of the 5 standard tests
    • SAS stands for side, angle, side
      If two sides and the angle between them can be shown to be the same, then the triangles are congruent
    • ASA stands for angle, side, angle
      If two angles and the side between them can be shown to be the same, then the triangles are congruent
    • AAS stands for angle, angle, side
      If two angles and one side can be shown to be the same, then the triangles are congruent
    • SSS stands for side, side, side
      If all three sides can be shown to be the same, then the triangles are congruent
    • RHS stands for right angle, hypotenuse, side
      Two right-angled triangles can be shown to be congruent if their hypotenuses (the longest side, opposite the right angle) and one other side can be shown to be the same

Congruent Triangles RN 1, downloadable IGCSE & GCSE Maths revision notes Congruent Triangles RN 2, downloadable IGCSE & GCSE Maths revision notes

Examiner Tip

  • To find equivalent angles or sides you may need to use a variety of skills
    • eg Parallel Lines, Symmetry, Isosceles Triangles, Circle Theorems, etc

Worked example

In the diagram below, A, B, C and D are four points on a circle.

X is the point of intersection of the lines AC and BD. 

M is the midpoint of AD.

XM is perpendicular to AD.
Congruent Triangles in a Circle, IGCSE & GCSE Maths revision notes

Prove that triangles AXB and DXC are congruent.
 

Congruent triangles have three equal angles and three equal sides.
You do not need to show that all of them are equal to prove congruence, you can look for one of the five standard tests of congruence. These are SAS, SSS, ASA, AAS, RHS.

Use the information given in the question to look for corresponding angles and sides that are equal.

Identify any corresponding sides that are equal between triangles AXB  and DXC.

As M is the midpoint of AD, AM = MD.
Triangles AMX and DMX  share the side MX, which meets AD at right angles so by the rule of congruency SAS, triangles AMX and DMX are congruent triangles.

 
The sides AX and DX are equal. 
Triangles AXM and CXM are congruent
 

Identify any corresponding angles that are equal between triangles AXB  and DXC.

Vertically opposite angles are equal.

Angle AXB = angle DXC 
Vertically opposite angles

Angles in the same segment are equal.

Angle ABX = angle DCX 
Angles subtended from the same arc are equal

The statement that if two corresponding angles and one side are the same then the two triangles are congruent must be made. This is the AAS property (angle, angle, side)

Write which angles or sides are equal out clearly in the proof, making sure to give the reasons why they are equal.

AX  = XD  as triangles AMX and DMX are congruent.
Angle AXB  = angle DXC as they are vertically opposite angles.
Angle ABX  = angle DCX as they are angles in the same segment.
 By the standard test AAS the triangles are congruent 

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.