Circle theorems

Geometry and measures

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A circle has centre $O$. Angle $AOB$ at the centre is $80°$, and $C$ is on the circumference. Find angle $ACB$, the angle at the circumference standing on the same arc.

2 marks
Triangle $ABC$ is drawn in a semicircle with $AB$ as diameter. Angle $BAC = 35°$. Find angle $ABC$.

2 marks
A tangent and a chord meet at $42°$. The chord and the tangent cut off an angle in the alternate segment. Find that angle.

3 marks
A cyclic quadrilateral has one angle of $110°$ and an adjacent angle of $80°$. Find the other two angles.

3 marks
$OA$ and $OB$ are radii, and angle $AOB = 100°$. Find the other two angles in triangle $AOB$.

4 marks
The radius meets a tangent at $T$. Explain why the angle between the radius and the tangent is $90°$, then find the third side of a right-angled triangle with $OT = 5$ cm and $OC = 13$ cm, where $C$ is a point on the tangent.

4 marks
$ABCD$ is a cyclic quadrilateral. Angle $ABC = 95°$ and angle $BAD = 70°$. Find angle $ADC$ and angle $BCD$.

5 marks
In the diagram, $O$ is the centre, angle $AOB = 140°$ and $TA$ is a tangent at $A$. Point $C$ is on the major arc $AB$. Find angle $ACB$, the base angles of triangle $AOB$, and the angle between the tangent and the chord $AB$.

Circle with centre O, angle AOB = 140°, and a tangent at A. Not to scale.

6 marks

Mark scheme

Answer. $40°$
- The angle at the centre is twice the angle at the circumference.
- $40°$
Answer. $55°$
- The angle in a semicircle is $90°$, so angle $C = 90°$.
- $180 - 90 - 35 = 55°$
Answer. $42°$
- The angle between a tangent and a chord equals the angle in the alternate segment.
- $42°$
Answer. $70°$ and $100°$
- Opposite angles in a cyclic quadrilateral sum to $180°$.
- The opposite of $110°$ is $70°$, and the opposite of $80°$ is $100°$.
Answer. $40°$ and $40°$
- $OA = OB$, so the triangle is isosceles.
- $(180 - 100) \div 2 = 40°$.
Answer. A radius is perpendicular to the tangent at the point of contact. $TC = 12$ cm.
- The radius is perpendicular to the tangent.
- $TC = \sqrt{13^{2} - 5^{2}} = 12$ cm.
Answer. $85°$ and $110°$
- Opposite angles sum to $180°$.
- Angle $ADC = 180 - 95 = 85°$.
- Angle $BCD = 180 - 70 = 110°$.
Answer. Angle $ACB = 70°$, base angles $20°$ each, and the tangent-chord angle is $70°$.
- The angle at the circumference is half of $140°$, so angle $ACB = 70°$.
- Triangle $AOB$ is isosceles, so each base angle is $\dfrac{180 - 140}{2} = 20°$.
- The angle between tangent $TA$ and chord $AB$ equals the angle in the alternate segment, $70°$.