Congruence (Edexcel IGCSE Maths)

Spec point — 6J

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

  • If all the angles are the same, then the shapes are similar

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

Exam Hint 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*.*

Answer:

Shapes C and D are congruent to A

Congruent Triangles

Spec point — 6I

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

Exam Hint AAA and SSA are not congruence conditions.

• AAA (all three angles the same) shows that the triangles are similar, but is not enough to show that they are congruent

• SSA (two sides and an angle not between the sides) is also not enough to prove congruence

  • Two triangles can meet the SSA condition without being congruent

Exam Hint The course specification does not explicitly mention the AAS test. It has however been accepted in past paper mark schemes for proving congruence.

• AAS and ASA are essentially equivalent

  • Because if you know two angles, you can always find the third one using 180° in a triangle
  • That converts 'AAS' to 'ASA'

Worked Example Prove that triangles ABC and PQR are congruent.

Answer:

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

Final answer: Triangles are congruent by the ASA condition