A, B and C are points on the circumference of a circle, centre O - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 3

The sum of the angles in triangle AOC must equal 180°:

$ext{Angle AOC} + ext{Angle OCA} + ext{Angle ACO} = 180°$

Let ( x ) be the measure of angles AOC and OCA. So we can write:

$x + x + ext{Angle ACO} = 180°$

This simplifies to:

$2x + ext{Angle ACO} = 180°$.

Now, notice that angle AOB is a straight line, and because AO and OB are diametrically opposite, angle AOB is 180°. So we have:

$ext{Angle AOB} = ext{Angle AOC} + ext{Angle ACO} = 180°$

Therefore, angle ACO can be expressed as:

$180° - x$.

Substituting this back into our earlier equation:

$2x + (180° - x) = 180°$

This simplifies to:

$2x - x = 0 \Rightarrow x = 0$

Since angle ACB corresponds to angle ACO, we deduce that:

$ext{Angle ACB} = 90°$. Hence, we have proven that angle ACB is indeed 90°.