Geometric proof

A geometric proof shows a fact is always true, one step at a time, with a reason for every step. Reasons are angle facts, properties of shapes, congruence conditions or circle theorems.

Key facts

  • Write each statement with its reason.
  • Useful reasons: angles on a line, vertically opposite, alternate, corresponding, co-interior, angles in a triangle, isosceles base angles.
  • End by saying what you have proved.

Worked examples

Example 1

Prove that the angles in a triangle add up to 180°.

  1. Draw a line through the top corner parallel to the base.
  2. The angles either side of the top angle equal the base angles a and b (alternate angles).
  3. Those three angles lie on a straight line, so they add to 180° (angles on a line). a + b + c = 180°

Answer: a + b + c = 180°

Example 2

Prove that an exterior angle of a triangle equals the sum of the two opposite interior angles.

  1. a + b + c = 180° (angles in a triangle).
  2. c + d = 180° (angles on a line).
  3. So d = a + b.

Answer: d = a + b