21.4 Proof of circle theorems — Practise with help

1. The opposite angles are equal.
2. $360^\circ$.

Start with the angle at the centre on a diameter.

The central angle is $180^\circ$. The circumference angle is half of it.

The diameter subtends a straight angle of $180^\circ$ at the centre. The angle at the circumference is half the centre angle, so it is $90^\circ$.

Compare triangles $OMA$ and $OMB$.

Use equal radii, the shared side $OM$, and the right angles at $M$.

$OA=OB$ because both are radii. $OM$ is shared, and both triangles are right-angled at $M$. The triangles are congruent by RHS, so corresponding sides $AM$ and $MB$ are equal.

Choose between the triangle angle sum and the cyclic quadrilateral theorem.

Opposite angles total $180^\circ$, so write $x+116=180$.

$x=180-116=64^\circ$. The cyclic quadrilateral theorem applies because the angles are opposite and all four vertices lie on one circle.