Circle theorems GCSE lesson: all eight on a grid, with the exact reasons to write

Circle theorems: practice

Give each angle as a number of degrees. Write the reason for each step on paper. The stretch is optional.

Diagnostic question: choose and explain

1. O is the centre of a circle. A, B and P are on the circle, with P on the major arc AB. Angle AOB = 110°. What is angle APB?
   1. 70°
   2. 220°
   3. 110°
   4. 55°

Practice questions

1. AB is a diameter of a circle, and C is another point on the circle. Angle CAB = 34°. Find angle ABC.
2. Which theorem answers each situation? O is the centre of the circle each time. Put each one in the group of the theorem you would use.They sort 10 items into Angle at the centre is twice the angle at the circumference and Angles in the same segment are equal and The angle in a semicircle is 90° and Opposite angles of a cyclic quadrilateral add up to 180° and A tangent is perpendicular to the radius.
3. The tangent at A and the chord AB make an angle of 70°. C is on the circle, in the other segment, and angle ACB = 70°. Which reason earns the mark?
   1. The alternate segment theorem
   2. Alternate angles are equal
   3. Angles in the same segment are equal
   4. Tangents from a point are equal
4. A and B are on a circle with centre O. Why is angle OAB equal to angle OBA?
   1. Angles in the same segment are equal
   2. OA and OB are both radii, so triangle OAB is isosceles
   3. The angle at the centre is twice the angle at the circumference
   4. Alternate angles are equal
5. The circle x² + y² = 25 has its centre at (0, 0). The tangents at (−5, 0) and at (0, 5) meet at a point T. Move the point to T.They move a point on a grid.
6. TA and TB are tangents from a point T, touching the circle at A and B. Angle ATB = 48°. Find angle TAB.
7. ABCD is a cyclic quadrilateral. Angle A = 3x + 15 and angle C = 2x − 10, in degrees. They are opposite angles, so they add up to 180°. Solve the equation line by line to find x. Write the numbers without the ° sign.They solve 3x + 15 + 2x − 10 = 180 line by line, each line checked as they write it.
8. O is the centre. A, B and C are on the circle, with C on the major arc AB. Angle AOB = 3x + 10 and angle ACB = x + 20, in degrees. Find x.
9. The tangent at A makes an angle of 64° with the chord AB. C is on the circle, in the other segment. Angle ABC = 52°. Find angle BAC.
10. A circle with centre O has radius 13 cm. The chord AB is 24 cm long. M is the point where the perpendicular from O meets AB. Find the distance OM, in cm.
11. Stretch: the tangent to x² + y² = 25 at (3, 4) is perpendicular to the radius from (0, 0) to (3, 4). Move the point to where the tangent crosses the line x = 7.They move a point on a grid.

Support: For the sorting question: for each situation, find the corners first. Is one at the centre? Are four on the circle? Is there a diameter, or a line touching the circle?

Circle theorems: exit

Find the angle and write a reason for each step, as a full sentence.

1. TA is a tangent to a circle with centre O, touching it at A. Angle ATO = 28°. Find angle AOT, and write a reason for each step.

Exit check

One angle found through two facts, each reason written, shows who can do both.

Students answer the exit question on their own: one angle, two reasons.

Circle theorems: angles and reasons

Find each angle or letter, and write a reason for every step as a full sentence (for example, "angles in the same segment are equal"). O is always the centre.