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SUBJECT NAME
Circle Theorem
9
1. The diagram shows points 𝐴, 𝐵, 𝐶 and 𝐷 on the circumference of a circle, centre 𝑂.
𝐴𝐷 is a straight line, 𝐴𝐵 = 𝐵𝐶 and angle 𝑂𝐴𝐵 = 52°.
Find angle 𝐴𝐷𝐶.
Angle 𝐴𝐷𝐶 =……………………………… °
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SUBJECT NAME
Circle Theorem
2. The diagram shows points 𝑃, 𝑄, 𝑅 and 𝑆 on the circumference of a circle, centre 𝑂.
𝑉𝑇 is the tangent to the circle at 𝑄.
Complete the statements.
(a) Angle 𝑄𝑃𝑆 = angle 𝑄𝑅𝑆 = ……………° because ………………………………………
…………………………………………………………………………………………….
(b) Angle 𝑆𝑄𝑃 = ……………° because ……………………………………………………
…………………………………………………………………………………………..
(c) Part (a) and (b) show that
the cyclic quadrilateral 𝑃𝑄𝑅𝑆 is a ………………………………
Page 2
SUBJECT NAME
Circle Theorem
3. (a)
𝐴, 𝐵, 𝐶 and 𝐷 are points on the circle. 𝐴𝐷 is parallel to 𝐵𝐶. The chords 𝐴𝐶 and 𝐵𝐷 intersect
at 𝑋. Find the value of 𝑢 and the value of 𝑣.
𝑢 = ………………………………
𝑣 = ………………………………
(b)
𝐹, 𝐺 and 𝐻 are points on the circle, centre 𝑂. Find the value of 𝑝.
𝑝 = ………………………………
Page 3
SUBJECT NAME
Circle Theorem
4.
𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 lie on the circle.
𝐴𝐵 is extended to 𝐹.
Angle 𝐴𝐸𝐷 = 140° and angle 𝐶𝐵𝐹 = 95°.
Find the values of 𝑤, 𝑥 and 𝑦.
𝑤 = ………………………………
𝑥 = ………………………………
𝑦 = ………………………………
Page 4
SUBJECT NAME
Circle Theorem
5. (a)
The points 𝐴, 𝐵, 𝐶 and 𝐷 lies on the circumference of the circle.
𝐴𝐵 = 𝐴𝐷, 𝐴𝐶 = 𝐵𝐶 and angle 𝐴𝐵𝐷 = 42°.
Find angle 𝐶𝐴𝐵.
Angle 𝐶𝐴𝐵 =……………………………… °
(b)
The points 𝑃, 𝑄, 𝑅 and 𝑆 lie on the circumference of the circle, centre 𝑂.
Angle 𝑄𝑂𝑆 = 146°. Find angle 𝑄𝑅𝑆.
Angle 𝑄𝑅𝑆 =……………………………… °
Page 5
SUBJECT NAME
Circle Theorem
6.
𝐴, 𝐵, 𝐶 and 𝐷 are points on the circle, centre 𝑂.
𝐵𝐶𝐸 is a straight line.
Angle 𝐴𝑂𝐶 = 108° and angle 𝐷𝐶𝐸 = 60°.
Calculate the values of 𝑤, 𝑥 and 𝑦.
𝑤 = ………………………………
𝑥 = ………………………………
𝑦 = ………………………………
Page 6
SUBJECT NAME
Circle Theorem
7.
In the diagram, points 𝐴, 𝐵, 𝐶, 𝐷, 𝐸 and 𝐹 lie on the circumference of the circle.
Angle 𝐵𝐹𝐶 = 19°, angle 𝐴𝐷𝐵 = 23° and angle 𝐴𝐵𝐸 = 67°.
Work out
(a) angle 𝐵𝐸𝐶,
Angle 𝐵𝐸𝐶 =……………………………… °
(b) angle 𝐴𝐵𝐶,
Angle 𝐴𝐵𝐶 =……………………………… °
(c) angle 𝐵𝐶𝐸.
Angle 𝐵𝐶𝐸 =……………………………… °
Page 7
SUBJECT NAME
Circle Theorem
8.(a)
𝑃, 𝑄 and 𝑅 lie on a circle, centre 𝑂.
Angle 𝑃𝑄𝑅 = 𝑦° and angle 𝑃𝑂𝑅 = (2𝑦 − 60)°.
Find the value of 𝑦.
𝑦 =………………………………
(b)
The points 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 lie on the circumference of the circle.
Angle 𝐷𝐶𝐸 = 47° and angle 𝐶𝐸𝐴 = 85°.
Find the values of 𝑤, 𝑥 and 𝑦.
𝑤 = ………………………………
𝑥 = ………………………………
𝑦 = ………………………………
Page 8
SUBJECT NAME
Circle Theorem
9. (a)
𝐴, 𝐵, 𝐶 and 𝐷 are points on the circumference of the circle, centre 𝑂.
𝐷𝑂𝐵 is a straight line and the angle 𝐷𝐴𝐶 = 58°.
Find angle 𝐶𝐷𝐵.
Angle 𝐶𝐷𝐵 =……………………………… °
(b)
𝑃, 𝑄 and 𝑅 are points on the circumference of the circle, centre 𝑂.
𝑃𝑂 is parallel to 𝑄𝑅 and angle 𝑃𝑂𝑄 = 48°.
Find angle 𝑂𝑃𝑅.
Angle 𝑂𝑃𝑅 =……………………………… °
Page 9
SUBJECT NAME
Circle Theorem
10.
𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 lie on the circle, centre 𝑂.
Angle 𝐴𝐸𝐵 = 35°, angle 𝑂𝐷𝐸 = 28° and angle 𝐴𝐶𝐷 = 109°.
(a) Work out the following angles, giving reasons for your answers.
(i) Angle 𝐸𝐵𝐷 = ……………….° because ………………………………………………
…………………………………………………………………………………………..
(ii) Angle 𝐸𝐴𝐷 = ……………….° because ………………………………………………
…………………………………………………………………………………………..
(b) Work out the angle 𝐵𝐸𝑂.
Angle 𝐵𝐸𝑂 =……………………………… °
Page 10
SUBJECT NAME
Circle Theorem
11.
𝐴, 𝐵, 𝐶 and 𝐷 lie on the circle.
𝑃𝐶𝑄 is a tangent to the circle at 𝐶.
Angle 𝐴𝐶𝑄 = 64°.
Work out angle 𝐴𝐵𝐶, giving reasons for your answer.
Angle 𝐵𝐴𝐶 =……………. because …………………………………………………………….
……………………………………………………………………………………………………
……………………………………………………………………………………………………
Page 11
