Hard GCSE Circle theorems Questions

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.

Set up the proof with the radii and the tangent that make it work.

$OA$ and $OB$ are radii drawn to the points of contact, so $\angle OAP = \angle OBP = 90^\circ$.

Follow the chain through to the expression asked for.

$OAPB$ is a quadrilateral, so its four angles add to $360^\circ$. Taking away the two right angles leaves $\angle AOB + x = 180$, so $\angle AOB = 180 - x$ degrees.

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 a tangent is perpendicular to the radius at the point of contact.

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.

The right angle is between the tangent and the RADIUS drawn to the point of contact, not to any other point of the circle.

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 option.

The correct option is $180 - x$.

Set up the proof with the radii and the tangent that make it work.

Each angle at the centre is twice the angle at the circumference standing on the same chord $BD$, which is where the $2x^\circ$ and the $2y^\circ$ come from.

Follow the chain through to the expression asked for.

The two angles $BOD$ are the angles at the point $O$, so they add to $360^\circ$: $2x + 2y = 360$. Dividing by $2$ gives $x + y = 180$, which is the cyclic quadrilateral theorem.

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 and 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.

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 option.

The correct option is $2x + 2y = 360$.

Set up the proof with the radii and the tangent that make it work.

The tangent is perpendicular to the diameter $AD$ at $A$, so $\angle DAT = 90^\circ$ and therefore $\angle DAB = 90 - x$ degrees.

Follow the chain through to the expression asked for.

$AD$ is a diameter, so $\angle ABD = 90^\circ$ (the angle in a semicircle). The angle sum of triangle $ABD$ then gives $\angle ADB = 180 - 90 - (90 - x) = x$ degrees — the tangent-chord angle, which is the alternate segment theorem.

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 a tangent is perpendicular to the radius at the point of contact and the angle in a semicircle is a right angle.

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.

The right angle is between the tangent and the RADIUS drawn to the point of contact, not to any other point of the circle.

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 option.

The correct option is $x$.

Use the angle in a semicircle.

$AB$ is a diameter and $C$ is on the circle, so the angle at $C$ is the angle in a semicircle and is a right angle.

Form an equation from the angle sum of the triangle.

The angles of triangle $ABC$ add to $180^\circ$. One of them is $90$, so the other two must add to $90$.

Collect like terms.

The $x$ terms add to $5x$ and the numbers add to $5$, so $5x = 85$.

Solve for x.

Dividing both sides by $5$ gives $x = 17$.

Substitute the value back into both angles.

Putting $x = 17$ back in gives $53^\circ$ and $37^\circ$.

Check the two acute angles add to a right angle.

$53 + 37 = 90$, which is what a right-angled triangle demands, so the value of $x$ is right.

Write down everything the question gives you before touching the theorems.

Getting the given angles onto the diagram first is what makes the right theorem obvious. Here you are told $\angle CAB = 53^\circ$ and $\angle ABC = 37^\circ$.

Recall that every radius of this circle is the same length.

Every point of a circle is the same distance from the centre, so $OA = OB = OC = r$. Any triangle with two radii as sides is therefore isosceles, and that is what turns a centre angle into two equal base angles.

Write down the angle sum of a triangle.

The three angles of any triangle add up to $180^\circ$. Most circle-theorem chains finish with this fact, so it is worth writing down early.

Write down the angle sum of a quadrilateral.

The four angles of any quadrilateral add up to $360^\circ$. This is the fact behind the cyclic quadrilateral theorem and behind the two-tangent kite.

Write down the angles at a point.

The angles round a single point add up to $360^\circ$. This is what links an angle at the centre to the reflex angle beside it.

Check that the answer is a possible size for this angle.

A non-reflex angle must lie between $0^\circ$ and $180^\circ$, and $53^\circ$ does, so the answer is at least possible. An answer outside that range would be wrong whatever the working said.

Name the theorem used, because the reason itself earns a mark.

The reason here is that the angle in a semicircle is a right angle. In a circle-theorem question the reason is worth as much as the number, so write it out in words.

Note the mistake this question is designed to catch.

The right angle is at the point on the circumference, not at the ends of the diameter.

State the value of x.

The value of $x$ is $17$.