1

$A$, $B$, $C$ and $D$ lie on a circle, forming a cyclic quadrilateral $ABCD$. Angle $A = 4x$ and the opposite angle $C = 5x$. Work out the size of angle $A$. [3]

2

$A$, $B$ and $C$ are points on a circle, centre $O$. The angle $ABC$ at the circumference is $2x+5$. The angle $AOC$ at the centre, standing on the same arc $AC$, is $5x-20$. Work out the size of angle $ABC$. [4]

3

$A$, $B$ and $C$ are points on a circle. The tangent to the circle at $A$ makes an angle of $3x-10$ with the chord $AB$. By the alternate segment theorem this equals the angle $ACB$ in the alternate segment, which is $2x+20$. Work out the size of angle $ACB$. [3]

4

$A$ and $B$ are points on a circle, centre $O$. $OA$ and $OB$ are radii, and the angle $AOB$ at the centre is $2x$. The angle $OAB$ is $x+24$. Work out the value of $x$, and hence the size of angle $AOB$. [5]

5

$ABCD$ is a cyclic quadrilateral. Angle $DAB = 2x+10$, angle $ABC = 3x$ and angle $BCD = x+50$. Work out the size of angle $ADC$. [5]