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Circle Theorems · MCQ assessment
Version A
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 48^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
ABase angles of an isosceles triangle are equalBAngles in the same segment are equalCThe angle at the centre is twice the angle at the circumferenceDThe angle in a semicircle is 90°
2$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 56^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AA tangent meets a radius at 90°BThe two tangents from a point are equal in lengthCOpposite angles of a cyclic quadrilateral add up to 180°DThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)
3$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 28^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
ABase angles of an isosceles triangle are equalBThe angle at the centre is twice the angle at the circumferenceCOpposite angles of a cyclic quadrilateral add up to 180°DThe angle in a semicircle is 90°
4In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $43^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)CAngles in the same segment are equalDA tangent meets a radius at 90°
5$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle CDA $= 74^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AAngles in the same segment are equalBA tangent meets a radius at 90°CThe angle at the centre is twice the angle at the circumferenceDOpposite angles of a cyclic quadrilateral add up to 180°
6$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 42^\circ$. Work out the size of angle CAB, marked $x$.
A48B90C138D42
7In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 50^\circ$. Work out the size of angle $x$.
A140B100C50D130
8$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 46^\circ$. Work out the size of angle $x$.
A44B46C134D90
9$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 108^\circ$. Work out the size of angle BCD, marked $x$.
A108B54C72D82
10$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 32^\circ$. Work out the size of angle CAB, marked $x$.
A148B32C58D90
11$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 41^\circ$. Work out the size of angle $x$.
A41B90C139D49
12In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. Angle $ABT$ is $40^\circ$. Work out the size of the angle between the tangent and the chord $TA$, marked $x$.
A80B40C90D140
13From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 62^\circ$. Work out the size of angle $PAB$, marked $x$.
A59B62C90D118
14$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. $AB$ is a diameter. Angle $CAB = 27^\circ$. Work out the size of angle $ABC$.
A27B63C117D153
15$PA$ and $PB$ are tangents to the circle, centre $O$, at $A$ and $B$. $C$ is a point on the circle. Angle $APB = 72^\circ$. Work out the size of angle $AOB$.
A72B288C108D162
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Circle Theorems · MCQ assessment
Version B
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 40^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BThe angle at the centre is twice the angle at the circumferenceCThe two tangents from a point are equal in lengthDAngles in the same segment are equal
2$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 52^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBThe angle in a semicircle is 90°COpposite angles of a cyclic quadrilateral add up to 180°DAngles in a triangle add up to 180°
3$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 62^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)BA tangent meets a radius at 90°CThe two tangents from a point are equal in lengthDThe angle in a semicircle is 90°
4$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle CDA $= 76^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AAngles in the same segment are equalBA tangent meets a radius at 90°COpposite angles of a cyclic quadrilateral add up to 180°DThe angle at the centre is twice the angle at the circumference
5From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 52^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)CThe angle in a semicircle is 90°DThe two tangents from a point are equal in length
6$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 72^\circ$. Work out the size of angle BCD, marked $x$.
A72B108C118D36
7$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 40^\circ$. Work out the size of angle $x$.
A50B140C90D40
8In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 47^\circ$. Work out the size of angle $x$.
A94B133C47D137
9$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 62^\circ$. Work out the size of angle CAB, marked $x$.
A28B118C90D62
10$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle BCD $= 86^\circ$. Work out the size of angle DAB, marked $x$.
A104B43C94D86
11$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 48^\circ$. Work out the size of angle $x$.
A42B90C132D48
12$X$, $Y$ and $Z$ lie on a circle with centre $O$. The chords $YX$ and $YZ$ are equal in length. Prove that $OY$ bisects $\angle XOZ$.
A$OX = OZ$ (radii), $YX = YZ$ (given) and $OY$ is common, so triangles $OYX$ and $OYZ$ are congruent (RHS). Hence $\angle XOY = \angle ZOY$, so $OY$ bisects $\angle XOZ$.B$OX = YX$ (radii), $YX = YZ$ (given) and $OY$ is common, so triangles $OYX$ and $OYZ$ are congruent (SSS). Hence $\angle XOY = \angle ZOY$, so $OY$ bisects $\angle XOZ$.C$OX = OZ$ (radii), $OY = YX$ (given) and $OY$ is common, so triangles $OYX$ and $OYZ$ are congruent (SSS). Hence $\angle XOY = \angle ZOY$, so $OY$ bisects $\angle XOZ$.D$OX = OZ$ (radii), $YX = YZ$ (given) and $OY$ is common, so triangles $OYX$ and $OYZ$ are congruent (SSS). Hence $\angle XOY = \angle ZOY$, so $OY$ bisects $\angle XOZ$.
13$A$, $B$, $C$ and $D$ are points on a circle. The chords $AC$ and $BD$ meet at $E$. Angle $BAC = 37^\circ$ and angle $ACD = 27^\circ$. Work out the size of angle $ABD$.
A63B37C27D153
14$PA$ and $PB$ are tangents to the circle, centre $O$, at $A$ and $B$. $C$ is a point on the circle. Angle $APB = 74^\circ$. Work out the size of angle $AOB$.
A164B106C286D74
15$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. $AB$ is a diameter. Angle $CAB = 51^\circ$. Work out the size of angle $ABC$.
A129B39C141D51
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Circle Theorems · MCQ assessment
Version C
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 29^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BAngles in the same segment are equalCA tangent meets a radius at 90°DThe angle at the centre is twice the angle at the circumference
2In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. The angle at the centre $AOB = 84^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle in a semicircle is 90°BAngles in the same segment are equalCAngles in a triangle add up to 180°DThe angle at the centre is twice the angle at the circumference
3From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 52^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBAngles in the same segment are equalCThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)DThe two tangents from a point are equal in length
4$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 50^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)BAngles in the same segment are equalCThe two tangents from a point are equal in lengthDA tangent meets a radius at 90°
5$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 39^\circ$. Work out the size of angle $ADB$, marked $x$.
A78B39C90D141
6$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 56^\circ$. Work out the size of angle $x$.
A34B124C56D90
7$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle CDA $= 100^\circ$. Work out the size of angle ABC, marked $x$.
A80B50C90D100
8$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 37^\circ$. Work out the size of angle CBA, marked $x$.
A37B143C53D90
9In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. The angle at the centre $AOB = 122^\circ$. Work out the size of angle $x$.
A122B29C58D61
10$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 34^\circ$. Work out the size of angle $x$.
A90B34C146D56
11$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle ABC $= 74^\circ$. Work out the size of angle CDA, marked $x$.
A106B116C37D74
12In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $68^\circ$. Work out the size of angle $ABT$, marked $x$.
A112B90C68D136
13$L$, $M$ and $N$ are points on a circle with centre $O$. Triangle $LMN$ is equilateral. Exactly one of the statements below must be true. Which one is it?
14$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 47^\circ$. Work out the size of the angle $AOB$ at the centre.
A94B86C133D47
15$A$, $B$, $C$ and $D$ are points on a circle. The chords $AC$ and $BD$ meet at $E$. Angle $BAC = 30^\circ$ and angle $ACD = 27^\circ$. Work out the size of angle $ABD$.
A153B63C27D30
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Circle Theorems · MCQ assessment
Version D
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 56^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe two tangents from a point are equal in lengthBA tangent meets a radius at 90°COpposite angles of a cyclic quadrilateral add up to 180°DThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)
2$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 56^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBThe angle in a semicircle is 90°COpposite angles of a cyclic quadrilateral add up to 180°DThe two tangents from a point are equal in length
3$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 92^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBAngles in the same segment are equalCOpposite angles of a cyclic quadrilateral add up to 180°DAngles in a triangle add up to 180°
4In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 47^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle in a semicircle is 90°BThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)CAngles in the same segment are equalDThe angle at the centre is twice the angle at the circumference
5$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 29^\circ$. Work out the size of angle $ADB$, marked $x$.
A151B58C29D90
6$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 28^\circ$. Work out the size of angle CAB, marked $x$.
A28B90C62D152
7$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle CDA $= 94^\circ$. Work out the size of angle ABC, marked $x$.
A47B96C94D86
8In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. The angle at the centre $AOB = 88^\circ$. Work out the size of angle $x$.
A92B88C44D46
9$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 43^\circ$. Work out the size of angle $x$.
A137B47C90D43
10$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 42^\circ$. Work out the size of angle CBA, marked $x$.
A138B48C90D42
11$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 84^\circ$. Work out the size of angle BCD, marked $x$.
A96B84C42D106
12In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $59^\circ$. Work out the size of angle $ABT$, marked $x$.
A90B118C121D59
13$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 42^\circ$. Work out the size of the angle $AOB$ at the centre.
A84B138C96D42
14$PA$ and $PB$ are tangents to the circle, centre $O$, at $A$ and $B$. $C$ is a point on the circle. Angle $APB = 64^\circ$. Work out the size of angle $AOB$.
A116B296C64D154
15$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. $AB$ is a diameter. Angle $CAB = 37^\circ$. Work out the size of angle $ABC$.
A143B53C37D127
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Circle Theorems · MCQ assessment
Version E
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 30^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe two tangents from a point are equal in lengthBThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)CA tangent meets a radius at 90°DAngles in the same segment are equal
2$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle ABC $= 108^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BThe angle at the centre is twice the angle at the circumferenceCThe two tangents from a point are equal in lengthDAngles in the same segment are equal
3$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 41^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BAngles in the same segment are equalCThe angle at the centre is twice the angle at the circumferenceDBase angles of an isosceles triangle are equal
4From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 42^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle in a semicircle is 90°BThe two tangents from a point are equal in lengthCThe angle at the centre is twice the angle at the circumferenceDThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)
5In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $46^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)BBase angles of an isosceles triangle are equalCAngles in the same segment are equalDA tangent meets a radius at 90°
6$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 74^\circ$. Work out the size of angle BCD, marked $x$.
A74B116C37D106
7$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 57^\circ$. Work out the size of angle $x$.
A123B90C33D57
8In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 55^\circ$. Work out the size of angle $x$.
A110B145C55D125
9$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 52^\circ$. Work out the size of angle CAB, marked $x$.
A38B90C128D52
10$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle ABC $= 114^\circ$. Work out the size of angle CDA, marked $x$.
A57B66C114D76
11$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 32^\circ$. Work out the size of angle $x$.
A148B32C90D58
12From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 44^\circ$. Work out the size of angle $PAB$, marked $x$.
A68B136C44D90
13$L$, $M$ and $N$ are points on a circle with centre $O$. Triangle $LMN$ is equilateral. Exactly one of the statements below must be true. Which one is it?
AThe minor arc $LM$ is one quarter of the circumferenceBThe minor arc $LM$ is one third of the circumferenceCThe minor arc $LM$ is one sixth of the circumferenceDThe minor arc $LM$ is half of the circumference
14$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 40^\circ$. Work out the size of the angle $AOB$ at the centre.
A140B100C40D80
15$PA$ and $PB$ are tangents to the circle, centre $O$, at $A$ and $B$. $C$ is a point on the circle. Angle $APB = 62^\circ$. Work out the size of angle $AOB$.
A62B118C152D298
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Circle Theorems · MCQ assessment
Version F
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 46^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe two tangents from a point are equal in lengthBA tangent meets a radius at 90°COpposite angles of a cyclic quadrilateral add up to 180°DThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)
2$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 34^\circ$. Work out the size of angle $ADB$, marked $x$.
A146B34C90D68
3From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 70^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe two tangents from a point are equal in lengthBThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)COpposite angles of a cyclic quadrilateral add up to 180°DThe angle at the centre is twice the angle at the circumference
4$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 30^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBOpposite angles of a cyclic quadrilateral add up to 180°CThe angle in a semicircle is 90°DAngles in a triangle add up to 180°
5$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 58^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AAngles in the same segment are equalBOpposite angles of a cyclic quadrilateral add up to 180°CThe angle at the centre is twice the angle at the circumferenceDThe two tangents from a point are equal in length
6$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 48^\circ$. Work out the size of angle $x$.
A42B48C90D132
7$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle ABC $= 112^\circ$. Work out the size of angle CDA, marked $x$.
A78B56C68D112
8$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 61^\circ$. Work out the size of angle CAB, marked $x$.
A90B29C61D119
9In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 56^\circ$. Work out the size of angle $x$.
A112B146C124D56
10$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 35^\circ$. Work out the size of angle $x$.
A90B145C35D55
11$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 116^\circ$. Work out the size of angle BCD, marked $x$.
A58B74C64D116
12$P$, $Q$ and $R$ are points on a circle with centre $O$. Triangle $PQR$ is equilateral. Exactly one of the statements below must be true. Which one is it?
13$X$, $Y$ and $Z$ lie on a circle with centre $O$. The chords $YX$ and $YZ$ are equal in length. Prove that $OY$ bisects $\angle XYZ$.
A$OY$ is common, $OX = OZ$ (radii) and $YX = YZ$ (given), so triangles $OYX$ and $OYZ$ are congruent (SAS). Hence $\angle OYX = \angle OYZ$, so $OY$ bisects $\angle XYZ$.B$OY$ is common, $OX = OZ$ (radii) and $YX = YZ$ (given), so triangles $OYX$ and $OYZ$ are congruent (SSS). Hence $\angle OYX = \angle OYZ$, so $OY$ bisects $\angle XYZ$.C$OY$ is common, $OX = OZ$ (radii) and $YX = YZ$ (given), so triangles $OYX$ and $OYZ$ are congruent (SSS). Hence $\angle OXY = \angle OZY$, so $OY$ bisects $\angle XYZ$.D$OX$ is common, $OX = OZ$ (radii) and $YX = YZ$ (given), so triangles $OYX$ and $OYZ$ are congruent (SSS). Hence $\angle OYX = \angle OYZ$, so $OY$ bisects $\angle XYZ$.
14$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 64^\circ$. Work out the size of the angle $AOB$ at the centre.
A116B52C64D128
15$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. Angle $AOC = 102^\circ$. Work out the size of angle $ABC$, marked $x$.
A258B51C102D78
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Circle Theorems · MCQ assessment
Version G
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 65^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AAngles in a triangle add up to 180°BOpposite angles of a cyclic quadrilateral add up to 180°CThe angle at the centre is twice the angle at the circumferenceDAngles in the same segment are equal
2$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 32^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe two tangents from a point are equal in lengthBOpposite angles of a cyclic quadrilateral add up to 180°CThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)DA tangent meets a radius at 90°
3In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. Angle $ABT$ is $35^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AA tangent meets a radius at 90°BThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)CAngles in the same segment are equalDThe two tangents from a point are equal in length
4From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 50^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)BThe angle at the centre is twice the angle at the circumferenceCAngles in the same segment are equalDThe two tangents from a point are equal in length
5$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 70^\circ$. Work out the size of angle $ADB$, marked $x$.
A90B70C140D110
6$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 35^\circ$. Work out the size of angle CBA, marked $x$.
A145B35C55D90
7$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 28^\circ$. Work out the size of angle $x$.
A152B90C28D62
8$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle BCD $= 66^\circ$. Work out the size of angle DAB, marked $x$.
A33B124C114D66
9In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 56^\circ$. Work out the size of angle $x$.
A124B146C56D112
10$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 43^\circ$. Work out the size of angle CBA, marked $x$.
A47B90C43D137
11$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 58^\circ$. Work out the size of angle $x$.
A90B58C122D32
12In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $53^\circ$. Work out the size of angle $ABT$, marked $x$.
A127B90C53D106
13$X$, $Y$ and $Z$ are points on a circle with centre $O$. Triangle $XYZ$ is equilateral. Exactly one of the statements below must be true. Which one is it?
AThe minor arc $XY$ is one sixth of the circumferenceBThe minor arc $XY$ is one quarter of the circumferenceCThe minor arc $XY$ is half of the circumferenceDThe minor arc $XY$ is one third of the circumference
14$A$, $B$ and $C$ lie on a circle with centre $O$. The chords $BA$ and $BC$ are equal in length. Prove that $OB$ bisects $\angle ABC$.
A$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (SAS). Hence $\angle OBA = \angle OBC$, so $OB$ bisects $\angle ABC$.B$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (SSS). Hence $\angle OBA = \angle OBC$, so $OB$ bisects $\angle ABC$.C$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (RHS). Hence $\angle OBA = \angle OBC$, so $OB$ bisects $\angle ABC$.D$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (SSS). Hence $\angle OAB = \angle OCB$, so $OB$ bisects $\angle ABC$.
15$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 41^\circ$. Work out the size of the angle $AOB$ at the centre.
A139B82C98D41
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Circle Theorems · MCQ assessment
Version H
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 46^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AAngles in a triangle add up to 180°BThe angle in a semicircle is 90°CThe angle at the centre is twice the angle at the circumferenceDAngles in the same segment are equal
2$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 40^\circ$. Work out the size of angle $ADB$, marked $x$.
A140B80C40D90
3In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. Angle $ABT$ is $43^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AAngles in a triangle add up to 180°BA tangent meets a radius at 90°CThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)DAngles in the same segment are equal
4$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle ABC $= 106^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BThe angle at the centre is twice the angle at the circumferenceCAngles in the same segment are equalDThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)
5$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 36^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)BAngles in a triangle add up to 180°CThe two tangents from a point are equal in lengthDA tangent meets a radius at 90°
6$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 50^\circ$. Work out the size of angle $x$.
A40B50C90D130
7$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 40^\circ$. Work out the size of angle CBA, marked $x$.
A90B140C40D50
8In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 41^\circ$. Work out the size of angle $x$.
A139B131C82D41
9$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle BCD $= 118^\circ$. Work out the size of angle DAB, marked $x$.
A62B118C72D59
10$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 49^\circ$. Work out the size of angle $x$.
A90B41C131D49
11$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 58^\circ$. Work out the size of angle CBA, marked $x$.
A90B32C58D122
12In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $44^\circ$. Work out the size of angle $ABT$, marked $x$.
A90B88C136D44
13$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. $AB$ is a diameter. Angle $CAB = 48^\circ$. Work out the size of angle $ABC$.
A42B132C138D48
14$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. Angle $AOC = 122^\circ$. Work out the size of angle $ABC$, marked $x$.
A58B238C122D61
15$A$, $B$, $C$ and $D$ are points on a circle. The chords $AC$ and $BD$ meet at $E$. Angle $BAC = 25^\circ$ and angle $ACD = 48^\circ$. Work out the size of angle $ABD$.
A132B25C42D48
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Circle Theorems · MCQ assessment
Version I
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 59^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle in a semicircle is 90°BThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)CThe angle at the centre is twice the angle at the circumferenceDAngles in the same segment are equal
2$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 49^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe two tangents from a point are equal in lengthBA tangent meets a radius at 90°CAngles in the same segment are equalDThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)
3$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 62^\circ$. Work out the size of angle $ADB$, marked $x$.
A90B62C118D124
4$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 52^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
ABase angles of an isosceles triangle are equalBThe angle at the centre is twice the angle at the circumferenceCOpposite angles of a cyclic quadrilateral add up to 180°DThe angle in a semicircle is 90°
5$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 58^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
ABase angles of an isosceles triangle are equalBOpposite angles of a cyclic quadrilateral add up to 180°CAngles in the same segment are equalDThe angle at the centre is twice the angle at the circumference
6In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. The angle at the centre $AOB = 88^\circ$. Work out the size of angle $x$.
A88B44C92D46
7$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 39^\circ$. Work out the size of angle CBA, marked $x$.
A141B51C39D90
8$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle DAB $= 66^\circ$. Work out the size of angle BCD, marked $x$.
A33B124C66D114
9$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 50^\circ$. Work out the size of angle $x$.
A40B90C130D50
10In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. The angle at the centre $AOB = 110^\circ$. Work out the size of angle $x$.
A110B35C55D70
11$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CBA $= 32^\circ$. Work out the size of angle CAB, marked $x$.
A148B58C90D32
12From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 40^\circ$. Work out the size of angle $PAB$, marked $x$.
A40B90C70D140
13$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 42^\circ$. Work out the size of the angle $AOB$ at the centre.
A84B42C138D96
14$A$, $B$ and $C$ lie on a circle with centre $O$. The chords $BA$ and $BC$ are equal in length. Prove that $OB$ bisects $\angle AOC$.
A$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (SAS). Hence $\angle AOB = \angle COB$, so $OB$ bisects $\angle AOC$.B$OA = OC$ (radii), $OB$ is common and $OB = BA$ (given), so triangles $OBA$ and $OBC$ are congruent (SSS). Hence $\angle AOB = \angle COB$, so $OB$ bisects $\angle AOC$.C$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (SSS). Hence $\angle AOB = \angle COB$, so $OB$ bisects $\angle AOC$.D$OA = OC$ (radii), $OB$ is common and $BA = BC$ (given), so triangles $OBA$ and $OBC$ are congruent (SSS). Hence $\angle OAB = \angle OCB$, so $OB$ bisects $\angle AOC$.
15$A$, $B$, $C$ and $D$ are points on a circle, centre $O$. Angle $AOC = 134^\circ$. Work out the size of angle $ABC$, marked $x$.
A134B67C46D226
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Circle Theorems · MCQ assessment
Version J
Tick one box (A, B, C or D) for each question. 15 questions, 1 mark each.Name: __________ Class: ______
Mark bar · shade a box for each question you got right
123456789101112131415
/ 15
1$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 36^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BThe two tangents from a point are equal in lengthCThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)DA tangent meets a radius at 90°
2$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 29^\circ$. Work out the size of angle $ADB$, marked $x$.
A151B90C29D58
3$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle ABC $= 112^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe angle at the centre is twice the angle at the circumferenceBA tangent meets a radius at 90°CAngles in the same segment are equalDOpposite angles of a cyclic quadrilateral add up to 180°
4$A$, $B$, $C$ and $D$ are points on a circle. Angle $ACB = 33^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AOpposite angles of a cyclic quadrilateral add up to 180°BThe angle at the centre is twice the angle at the circumferenceCA tangent meets a radius at 90°DAngles in the same segment are equal
5From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 50^\circ$. Which circle theorem would you use to work out the size of the angle marked $x$?
AThe alternate segment theorem (tangent–chord angle equals the angle in the alternate segment)BOpposite angles of a cyclic quadrilateral add up to 180°CThe two tangents from a point are equal in lengthDThe angle at the centre is twice the angle at the circumference
6$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 57^\circ$. Work out the size of angle CBA, marked $x$.
A123B90C57D33
7$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TOP = 33^\circ$. Work out the size of angle $x$.
A90B33C57D147
8In the diagram, $A$, $B$ and $C$ are points on a circle, centre $O$. Angle $ACB = 44^\circ$. Work out the size of angle $x$.
A88B134C136D44
9$A$, $B$, $C$ and $D$ are points on a circle, forming a cyclic quadrilateral $ABCD$. Angle BCD $= 92^\circ$. Work out the size of angle DAB, marked $x$.
A46B98C88D92
10$A$, $B$ and $C$ are points on a circle and $AB$ is a diameter. Angle CAB $= 29^\circ$. Work out the size of angle CBA, marked $x$.
A151B61C90D29
11$T$ is a point on a circle, centre $O$. $TP$ is a tangent to the circle. Angle $TPO = 40^\circ$. Work out the size of angle $x$.
A40B90C140D50
12In the diagram, $T$ is a point on a circle and the straight line through $T$ is a tangent. $TA$ is a chord and $B$ is a point on the circle. The angle between the tangent and the chord $TA$ is $48^\circ$. Work out the size of angle $ABT$, marked $x$.
A132B96C48D90
13$A$, $B$ and $C$ are points on a circle with centre $O$. Triangle $ABC$ is equilateral. Exactly one of the statements below must be true. Which one is it?
14From a point $P$ outside a circle, centre $O$, two tangents are drawn touching the circle at $A$ and $B$. Angle $APB = 48^\circ$. Work out the size of angle $PAB$, marked $x$.
A66B48C132D90
15$A$, $B$ and $C$ are points on a circle, centre $O$. $OA$ and $OB$ are radii. Angle $OAB = 64^\circ$. Work out the size of the angle $AOB$ at the centre.
A64B52C116D128
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Circle Theorems