1fully worked

AB is a diameter. Angle CAB $=35°$. Find angle $x=$ angle CBA.

1AB is a diameter, so the angle in a semicircle is a right angleName the circle theorem

2angle ACB $=90°$ – mark it at CWrite the fact it gives you

3triangle ACB: $35°+90°+x=180°$Set up the calculation

4$x=180°-125°=55°$Work out $x$

Check - Why is angle ACB exactly $90°$?

A because AB is a diameter, and the angle in a semicircle is a right angleB because triangle ACB is equilateralC because $35°$ doubled is $70°$D because every angle in a circle is $90°$

2you finish it

$A$, $B$ and $C$ are points on a circle, centre $O$. Angle $AOB=96°$. Find angle $x=$ angle $ACB$.

1The angle at the centre is twice the angle at the circumferenceName the circle theorem

2angle $AOB$ (at the centre) and angle $x$ (at $C$) stand on the same arc $AB$Write the fact it gives you

3$x = 96° \div 2$Set up the calculation

4Work out $x$

Check - Angle $AOB$ is at the centre. How is it related to angle $x$ at the circumference?

A they are equalB they add up to $180°$C the centre angle is twice the circumference angleD the centre angle is half the circumference angle

3you finish it

$ABCD$ is a cyclic quadrilateral. Angle $DAB=98°$. Find angle $x=$ angle $BCD$.

1Opposite angles of a cyclic quadrilateral add up to $180°$Name the circle theorem

2angle $DAB$ and angle $BCD$ are OPPOSITE angles of the quadrilateralWrite the fact it gives you

3Set up the calculation

4Work out $x$

Check - Which pair of angles does the cyclic quadrilateral theorem link?

A angles next to each otherB angles opposite each otherC the two smallest anglesD any angle and the angle at the centre

4you do it all

$TP$ is a tangent touching the circle, centre $O$, at $T$. Angle $TPO=32°$. Find angle $x=$ angle $TOP$.

1Name the circle theorem

2Write the fact it gives you

3Set up the calculation

4Work out $x$

Check - Why is the angle at $T$ exactly $90°$?

A because triangle $OTP$ is isoscelesB because angles on a straight line add to $180°$C because $32°+58°=90°$D because a tangent is perpendicular to the radius at the point of contact