21.2 Circle theorems — Practise

Work independently. Questions become more demanding as you progress. Check your answers after you have attempted the set.

1. H An angle at the circumference is $34^\circ$. Find the angle at the centre standing on the same arc.
2. H An angle at the centre is $126^\circ$. Find the angle at the circumference standing on the same arc.
3. H $AB$ is a diameter and $C$ is on the circle. Find $\angle ACB$.
4. H Two angles stand on the same chord in the same segment. One is $48^\circ$. Find the other.
5. H Opposite angles in a cyclic quadrilateral are $x^\circ$ and $117^\circ$. Find $x$.
6. H Opposite angles in a cyclic quadrilateral are $(3x+5)^\circ$ and $(2x+25)^\circ$. Solve for $x$.
7. H An angle at the circumference is $(x+12)^\circ$. The angle at the centre on the same arc is $100^\circ$. Find $x$.
8. H A cyclic quadrilateral has one angle of $88^\circ$. Find the opposite angle.
9. H In a triangle inscribed in a semicircle, one acute angle is $37^\circ$. Find the third angle.
10. H A cyclic quadrilateral has three angles $72^\circ$, $105^\circ$ and $x^\circ$. The $105^\circ$ angle is opposite $x$. Find $x$.