The figure below shows a circle with centre O. Points A, B and C lie on the circle. Angle ABC = 100°. Find angle OAC, giving reasons for each step of your working.

1. Find the reflex angle at the centre subtended by chord $AC$.

$\angle ABC = 100^\circ$ is an angle at the circumference standing on arc $AC$.

Hence the reflex angle at the centre standing on the same arc is
$\text{reflex }\angle AOC = 2 \times 100^\circ = 200^\circ.$
Reason: angle at the centre is twice the angle at the circumference standing on the same arc.

2. Find the smaller angle $AOC$ inside triangle $AOC$.

$\angle AOC = 360^\circ - 200^\circ = 160^\circ.$

3. Use equal radii to identify triangle $AOC$ as isosceles.

Since $OA$ and $OC$ are radii of the same circle,
$OA = OC.$
So $\triangle AOC$ is isosceles and
$\angle OAC = \angle OCA.$

4. Calculate the base angle $OAC$.

Using the angle sum of a triangle,
$\angle OAC + \angle OCA + \angle AOC = 180^\circ.$
Since $\angle OAC = \angle OCA$,
$2\angle OAC + 160^\circ = 180^\circ,$
$2\angle OAC = 20^\circ,$
$\angle OAC = 10^\circ.$