1

$A$, $B$, $C$ and $D$ are four points on a circle, in that order, forming a cyclic quadrilateral $ABCD$. Angle $DAB = 4x+5$ and the opposite angle $BCD = x+65$. A student argues that opposite angles of a cyclic quadrilateral are always equal to each other, so writes $4x+5=x+65$ and gets $x=20$. State what is wrong with the student's method, then use the correct theorem to work out the true size of angle $DAB$. [3]

2

$P$ is a point outside a circle, centre $O$. $PA$ and $PB$ are tangents to the circle, touching it at $A$ and $B$. Because $PA=PB$, triangle $PAB$ is isosceles, so angle $PAB=$ angle $PBA$. Angle $APB = x^2+x+50$ and angle $PAB = 3x+5$. Work out the size of angle $APB$, rejecting any value of $x$ that does not give a valid triangle. [4]

3

$A$, $B$ and $C$ are points on a circle, centre $O$. The angle $ABC$ at the circumference is $25$ degrees. The angle $AOC$ at the centre, standing on the same arc $AC$, is $5x$, where $x$ is a positive integer. Work out the value of $x$. [3]

4

$A$, $B$, $C$ and $D$ are four points on a circle, in that order around the circle, and $AB$ is a diameter. $ABCD$ is a cyclic quadrilateral. Angle $CAB = x$ and angle $ADC = 4x-30$. Work out the size of angle $ABC$. [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]