Notifications

Log in to see your notifications.

Practise this topic

Congruence

9 min read

Same shape, same size

Two shapes are congruent when one can be moved onto the other exactly, by sliding, turning or flipping. Congruent shapes have equal matching sides and equal matching angles.

The triangle turns about the origin. Its size and shape do not change, so the two triangles are congruent.

Conditions for congruent triangles

You do not need all six measurements. One of these sets is enough.
TestWhat must match
SSSall three sides
SAStwo sides and the angle between them
ASAtwo angles and the side between them
RHSright angle, hypotenuse and one other side
Conditions for congruent trianglesFour triangles marked to show the four tests for congruence: SSS three sides equal, SAS two sides and the angle between them, ASA two angles and a side, and RHS a right angle with the hypotenuse and one other side.SSSSASASARHS
Try itSSS test

Triangle 1 has sides 5, 6 and 7. Change the sides of triangle 2. When every difference is zero the triangles are congruent by SSS.

Worked exampleProve two triangles congruent

In triangles ABC and PQR, AB = PQ = 5 cm, BC = QR = 8 cm and angle B = angle Q = 40°. Are the triangles congruent?

  1. AB = PQ (given): one pair of sides.
  2. Angle B = angle Q (given): the angle between those sides.
  3. BC = QR (given): the other side.

Answer: Yes, SAS: two sides and the angle between them are equal, so △ABC ≡ △PQR.

Check yourself Which test is NOT enough to prove triangles congruent?
Check yourself Triangles have sides 4, 5, 6 and 4, 5, 6. Which test shows they are congruent?

Key terms

Congruent
Exactly the same shape and size.
Hypotenuse
The longest side of a right-angled triangle.

Summary

  • Congruent shapes match exactly in size and shape.
  • Tests for triangles: SSS, SAS, ASA, RHS.
  • AAA is not a congruence test.
Logarithms Similarity