Geometric Proofs

A geometric proof is a step-by-step logical argument that shows a statement about a shape or angle is true for *all* possible cases. It’s a key higher-tier skill that uses established facts to build a solid case and prove a conclusion beyond doubt.

Your Proof Toolkit

Every proof is built upon a foundation of known facts. You must know these rules and be able to state them as reasons in your argument.

Angle Facts

Angles on a straight line sum to $180^\circ$.

Angles at a point sum to $360^\circ$.

Vertically opposite angles are equal.

Alternate angles (Z-shape) are equal.

Corresponding angles (F-shape) are equal.

Co-interior angles (C-shape) sum to $180^\circ$.

Triangle Congruence Criteria

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

RHS (Right-angle-Hypotenuse-Side)

Worked Example: Proving a Theorem

Prove that the sum of angles in a triangle is $180^\circ$.

Statement	Reason
Draw line PQ through A, parallel to BC.	Construction
$\angle PAB = \angle ABC = b$	Alternate angles are equal.
$\angle QAC = \angle ACB = c$	Alternate angles are equal.
$\angle PAB + \angle BAC + \angle QAC = 180^\circ$	Angles on a straight line sum to $180^\circ$.
Therefore, $b + a + c = 180^\circ$.	Substitution.

Tutor Insights

🤔 Common Misunderstandings

Examples are not proofs. Showing $3+5=8$ suggests a rule for odd numbers, but doesn’t prove it for all cases. Algebra is needed for that. There are a small amount of questions where a counter example can act as proof. For example, prove that the sum of two numbers can be even or odd. Show and example of both. But there are not many of these types of questions.
Assuming facts. You can only use information that is given or that you can deduce. Just because lines look parallel doesn’t mean they are unless marked.

📝 Common Exam Mistakes

Missing or incorrect reasons. This is the biggest cause of lost marks. You must justify every step.
Poorly structured proofs. A proof must be a clear, logical sequence of steps.
Not using correct geometric language and notation for angles, lines, and congruence. For example you can’t say z angles. You must say alternate angles.

Practice Questions

In the diagram, $AD$ and $BC$ intersect at $O$, and $AB$ is parallel to $DC$. Prove that $\triangle AOB$ is similar to $\triangle DOC$.

Show Answers

Proof:

Statement	Reason
$\angle AOB = \angle DOC$	Vertically opposite angles are equal.
$\angle OAB = \angle OCD$	Alternate angles are equal (since $AB \parallel DC$).
$\angle OBA = \angle ODC$	Alternate angles are equal (since $AB \parallel DC$).
Therefore, $\triangle AOB \sim \triangle DOC$	AAA (Angle-Angle-Angle) similarity.