Writing Geometric Proofs

Grade 6

What a proof needs

A geometric proof is a logical chain: each step follows from the one before, and every statement must be backed by a reason. The reason is the angle fact, theorem, or property that justifies your step.

Without reasons, your working is just a list of numbers — it earns far fewer marks. Every step: state the angle or fact, then give the reason in brackets or on the same line.

Structure to follow

1. Identify what you are trying to show. Write it down.

2. Start from what you know. Use angle facts step by step until you reach the target statement.

3. Finish with a clear conclusion: "Therefore..." or "Hence...". Do not leave the reader guessing what you proved.

Common reasons to have ready: "angles on a straight line (180°)", "vertically opposite angles (equal)", "alternate angles (equal, parallel lines)", "corresponding angles (equal, parallel lines)", "angles in a triangle (180°)", "base angles of an isosceles triangle (equal)".

Worked example

EXAMPLE
AB is parallel to CD. Prove that angle ABE + angle CEB = 180°, where E is a point between the lines.
Draw a line through E parallel to AB and CD. Angle ABE equals the alternate angle at E on the AB-side (alternate angles, parallel lines). Angle CEB equals the alternate angle at E on the CD-side (alternate angles, parallel lines). These two alternate angles sit on a straight line at E and sum to 180°. Therefore angle ABE + angle CEB = 180° (co-interior angles, parallel lines).
Nova
Try this one

Each interior angle of a regular polygon is \( 156^\circ \).

Work out how many sides the polygon has. [3]

Nova's hint:

This uses the link between interior and exterior angles, then the exterior-angle total.

Ask yourself: 'if I know the interior angle, how do I find the exterior angle, and how does that give the number of sides?' The exterior angles share \( 360^\circ \) equally.

Try to Solve it with Nova?