GCSE Circle theorems Practice Questions

Identify which angle is at the centre and which is at the circumference.

Both angles stand on the same chord $AB$. $O$ is the centre, so $\angle AOB = 140^\circ$ is the angle at the centre; $C$ is on the circumference, so $\angle ACB$ is the angle at the circumference.

Halve the angle at the centre.

The angle at the centre is twice the angle at the circumference, so the angle at the circumference is half the angle at the centre. Half of $140^\circ$ is $70^\circ$.

State the size of the angle ACB.

The angle $ACB$ is $70^\circ$.

Work out which theorem the configuration is built on.

The reason here is that the angle in a semicircle is a right angle.

Check that theorem really does produce the stated angle.

Applying it to the angles given produces $90^\circ$, which is exactly the value the question states, so this is the reason that works.

State the correct reason.

The angle in a semicircle is a right angle.

Work out which theorem the configuration is built on.

The reason here is that the angle between a tangent and a chord equals the angle in the alternate segment.

Check that theorem really does produce the stated angle.

Applying it to the angles given produces $54^\circ$, which is exactly the value the question states, so this is the reason that works.

Deal with every option, not just the one that looks right.

In a multiple-choice circle-theorem question the wrong options are the answers you get by using the wrong theorem. Working out what each one would need to be true is the fastest way to be sure.

Name the theorem that actually applies here.

The theorem in play is that the angle between a tangent and a chord equals the angle in the alternate segment.

Check the conditions of that theorem are met.

Every circle theorem has conditions: the points must be on the circumference, a tangent must touch at exactly one point, a diameter must pass through the centre. Check them before quoting the theorem.

State the correct reason.

The angle between a tangent and a chord is equal to the angle in the alternate segment.

Write down the test for a cyclic quadrilateral.

The angles are listed in order round the quadrilateral, so the first and third are opposite, and so are the second and fourth. Each opposite pair must add to $180^\circ$.

Apply the test to the correct option.

Both opposite pairs add to $180^\circ$, so these four angles really could belong to a cyclic quadrilateral.

Deal with every option, not just the one that looks right.

In a multiple-choice circle-theorem question the wrong options are the answers you get by using the wrong theorem. Working out what each one would need to be true is the fastest way to be sure.

Name the theorem that actually applies here.

The theorem in play is that opposite angles of a cyclic quadrilateral add up to $180^\circ$.

Check the conditions of that theorem are met.

Every circle theorem has conditions: the points must be on the circumference, a tangent must touch at exactly one point, a diameter must pass through the centre. Check them before quoting the theorem.

Note the mistake the wrong options are built from.

It is the OPPOSITE angles that add to $180^\circ$, not the adjacent ones. Adjacent angles of a cyclic quadrilateral are not related.

Write down the angle sum of a triangle.

The three angles of any triangle add up to $180^\circ$.

Write down the angle sum of a quadrilateral.

The four angles of any quadrilateral add up to $360^\circ$.

Write down the angles at a point.

The angles round a single point add up to $360^\circ$.

State the correct option.

The correct option is $85^\circ, 100^\circ, 95^\circ, 80^\circ$.

Work out which theorem the configuration is built on.

The reason here is that the angle at the centre is twice the angle at the circumference.

Check that theorem really does produce the stated angle.

Applying it to the angles given produces $65^\circ$, which is exactly the value the question states, so this is the reason that works.

Deal with every option, not just the one that looks right.

In a multiple-choice circle-theorem question the wrong options are the answers you get by using the wrong theorem. Working out what each one would need to be true is the fastest way to be sure.

Name the theorem that actually applies here.

The theorem in play is that the angle at the centre is twice the angle at the circumference.

Check the conditions of that theorem are met.

Every circle theorem has conditions: the points must be on the circumference, a tangent must touch at exactly one point, a diameter must pass through the centre. Check them before quoting the theorem.

Note the mistake the wrong options are built from.

Do not double when you should halve. The angle at the CENTRE is the big one; the angle at the CIRCUMFERENCE is half of it.

Write down the angle sum of a triangle.

The three angles of any triangle add up to $180^\circ$.

Write down the angle sum of a quadrilateral.

The four angles of any quadrilateral add up to $360^\circ$.

Write down the angles at a point.

The angles round a single point add up to $360^\circ$.

Remember that a diagram is never the evidence.

Circle-theorem diagrams are not drawn accurately. An option cannot be chosen because it looks about right on the page.

Say why the reason matters as much as the answer.

Even in a multiple-choice question, being able to name the theorem is what transfers to the written questions, where the reason carries the mark.

Check the size of the chosen angle is possible.

An angle at a point on a circle must be a sensible size. Any option outside the possible range can be discarded straight away.

Note that the radius of the circle is irrelevant.

None of the theorems mention the size of the circle. Scaling the whole picture changes no angle at all.

Set the reasoning out as a chain.

One angle, one value, one reason on each line. That is what the mark scheme rewards in the written version of this question.

State the correct reason.

The angle at the centre is twice the angle at the circumference.