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

Now, observe triangle AOB. The angles in this triangle must also sum to 180 degrees:

$ext{angle AOB} + ext{angle OAB} + ext{angle OBA} = 180°$

We already know that angle AOB is 180°, so:

$ext{angle OAB} + ext{angle OBA} = 180° - 180° = 0°$

This indicates that the angles at point O are such that they cannot be equal to the angles at point A and point B in triangle ABC.

Since we derived that angle OAB and angle OBA both add up and are supplementary to angle AOB, it implies that angles A and B of triangle ABC must also relate such that they reflect into this formation. Therefore:

$ext{angle ACB} + ext{(angle OAB)} + ext{(angle OBA)} = 180°$

Thus, since angle OAB and angle OBA equal zero, then:

$ext{angle ACB} = 90°$

This directly demonstrates that angle ACB is 90 degrees.