16.5 Geometrical reasoning and proof — Learn

Prerequisites: use angle facts from Sections 16.1–16.4 and solve linear equations.

Objectives: construct a clear geometrical argument, state the fact used at each step, and prove angle results. This Higher-only section covers G6.

Build an argument from known facts

A geometrical proof is a chain of statements where each step follows from a known fact or an earlier step. A reason names that fact.

Example 1. Two parallel lines are crossed by a transversal. One angle is $64^\circ$. Prove that the co-interior angle on the same side is $116^\circ$.

Let the unknown angle be $x$.

Example 2. Explain why the exterior angle of a triangle equals the sum of its opposite interior angles.

Let the opposite interior angles be $a$ and $b$, and the adjacent interior angle be $c$. The triangle sum gives $a+b+c=180^\circ$. The exterior angle $e$ and $c$ lie on a straight line, so $e+c=180^\circ$. Both sums equal $180^\circ$, so subtracting $c$ gives $e=a+b$.

Watch out. Writing only an answer does not show a proof. State the angle fact beside each equation, then show how the equations give the result.

Progress check

For the triangular courtyard tile, two interior angles are $68^\circ$ and $47^\circ$. Write one line that gives the remaining angle and its reason.

You should now be able to

• organise a proof as statements supported by reasons;
• use equal angles or angle sums in a multi-step argument;
• explain why an angle result follows, rather than giving only its value.