Circle theorems I: IGCSE Mathematics questions with answers
Free Circle theorems I questions for IGCSE Mathematics (0580), part of Geometry. Each question has the answer and the marking points an examiner would look for.
A, B, C and D lie on a circle in that order, so ABCD is a cyclic quadrilateral. TA is the tangent to the circle at A. Angle ABC = (3x + 20)°, angle ADC = (110 − x)° and angle TAB = 55°, where T is on the opposite side of AB from C.
(a)
Find the value of x. Give a reason for each step.
[3]
(b)
Find angle ACB. Give a reason.
[2]
(c)
Find angle BAC. Give reasons.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
x = 25
opposite angles of a cyclic quadrilateral add up to 180° B1 3x + 20 + 110 − x = 180 M1 x = 25 A1
3
(b)
55°
55° B1 alternate segment theorem B1
2
(c)
30°
angle ABC = 95° B1 angles in triangle ABC add to 180° M1 30° A1
3
Question 2Structured · 2 marks
A, B, C and D are points on the circle.
Fig. 2.1
Find the value of y. Give a reason for your answer.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
y = 29
29 B1 Angles in the same segment are equal B1
2
Question 3Structured · 8 marksA* challenge
A, B, C and D lie on a circle in that order, so ABCD is a cyclic quadrilateral. TA is the tangent to the circle at A. Angle ABC = (2x + 10)°, angle ADC = (150 − x)° and angle TAB = 55°, where T is on the opposite side of AB from C.
(a)
Find the value of x. Give a reason for each step.
[3]
(b)
Find angle ACB. Give a reason.
[2]
(c)
Find angle BAC. Give reasons.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
x = 20
opposite angles of a cyclic quadrilateral add up to 180° B1 2x + 10 + 150 − x = 180 M1 x = 20 A1
3
(b)
55°
55° B1 alternate segment theorem B1
2
(c)
75°
angle ABC = 50° B1 angles in triangle ABC add to 180° M1 75° A1
3
Question 4Structured · 2 marks
A, B, C and D are points on the circle. ABCD is a cyclic quadrilateral.
Fig. 4.1
Find the value of y. Give a reason for your answer.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
y = 109
109 B1 Opposite angles of a cyclic quadrilateral add up to 180° B1
2
Question 5Structured · 8 marksA* challenge
A, B, C and D lie on a circle in that order, so ABCD is a cyclic quadrilateral. TA is the tangent to the circle at A. Angle ABC = (3x + 10)°, angle ADC = (110 − x)° and angle TAB = 50°, where T is on the opposite side of AB from C.
(a)
Find the value of x. Give a reason for each step.
[3]
(b)
Find angle ACB. Give a reason.
[2]
(c)
Find angle BAC. Give reasons.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
x = 30
opposite angles of a cyclic quadrilateral add up to 180° B1 3x + 10 + 110 − x = 180 M1 x = 30 A1
3
(b)
50°
50° B1 alternate segment theorem B1
2
(c)
30°
angle ABC = 100° B1 angles in triangle ABC add to 180° M1 30° A1
3
Question 6Structured · 8 marksA* challenge
A, B, C and D lie on a circle in that order, so ABCD is a cyclic quadrilateral. TA is the tangent to the circle at A. Angle ABC = (3x + 10)°, angle ADC = (100 − x)° and angle TAB = 50°, where T is on the opposite side of AB from C.
(a)
Find the value of x. Give a reason for each step.
[3]
(b)
Find angle ACB. Give a reason.
[2]
(c)
Find angle BAC. Give reasons.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
x = 35
opposite angles of a cyclic quadrilateral add up to 180° B1 3x + 10 + 100 − x = 180 M1 x = 35 A1
3
(b)
50°
50° B1 alternate segment theorem B1
2
(c)
15°
angle ABC = 115° B1 angles in triangle ABC add to 180° M1 15° A1
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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in