Circle Theorems with reasons - Practice quiz.

A, B and C are points on the circumference of a circle, centre O.

Angle ABC = 43⁰

a) Find the size of the angle AOC

angle AOC = ​ (a)  

b) Give a reason for your answer:

​ ​ (b)  

• 86
• Center angle = twice the circumference angle.
• Angles in the same segment are equal.
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Tangents meet the radius at 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 43
• 90

B and D are points on the circumference of a circle, centre O

ABC is a tangent to the circle at B

a) Find the size of angle x

angle x = ​ ​ (a)  

b) Give a reason for your answer:

​ ​ ​ (b)   ​​

• 62
• Tangents meet the radius at 90 degrees.
• Angles in the same segment are equal.
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 28
• 90
• Center angle = twice the circumference angle

A, B, C and D are points on the circumference of a circle.

Angle BDC is 38⁰ and one of the Angles at the intersection of the Chords BD and AC is 86⁰ as indicated.

a) Find the size of angle BAC.

angle BAC = ​ ​ ​ (a)  

b) Give a reason for your answer:

​ ​ ​ (b)   ​​

• 38
• Angles in the same segment are equal.
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 76
• 48
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees

A, B, C and D are points on a circle.

Angle BAC = 40⁰

Angle DBC = 55°.

a) Find the size of the angle DAC.

angle DAC = ​ ​ ​ ​ (a)  

b) Give a reason for your answer:

​ ​ ​ (b)   ​​

• 55
• Angles in the same segment are equal.
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 40
• 95
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees

A, B, C and D are points on a circle, centre O. AE is a tangent to the circle. Angle ADC = 68°

a) Find the size of angle ABC.

angle ABC = ​ ​ ​ ​ (a)   ​

b) Give a reason for your answer:

​ ​ ​ ​ (b)  

• 112
• Opposite angles in a cyclic quad add to 180
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 136
• 68
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees

P, Q and R are points on the circumference of a circle, centre O.

PQ passes through O.

Angle RPQ = 67⁰.

a) Find the size of the angle PQR.

angle PQR = ​​ (a)   ​ ​ ​

b) Give a reason for your answer:

​ ​ ​ ​ (b)  

• 23
• The Angle in a semi-circle is always 90 degrees
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 67
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees
• 134

E, F, G and H are points on the circumference of a circle.

Angle HEF has a size of 99⁰.

a) Find the size of angle FGH.

angle FGH = ​ ​ ​ ​ (a)   ​

b) Give a reason for your answer:

​ ​ ​ ​ (b)  

• 81
• Opposite angles in a cyclic quad add to 180
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 80
• 50
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees

Q and P are points on the circumference of a circle, centre O

a) Find the size of the angle x.

angle x = ​​ (a)   ​ ​ ​

b) Give a reason for your answer:

​ ​ ​ ​ (b)  

• 26
• Triangle OQP is an Isosceles triangle.
• The angle in a semi-circle is always 90 degrees.
• Angles in the same segment are equal.
• Triangle OQP is an equilateral triangle.
• Tangents which meet at an external point are equal
• 52
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees
• 134

A, B, C and D are points on a circle, centre O. AE is a tangent to the circle. Angle ADC = 68°

a) Find the size of angle AOC.

angle AOC = ​ ​ ​ ​ (a)   ​

b) Give a reason for your answer:

​ ​ ​ ​ ​ (b)  

• 136
• Center angle = twice the circumference angle
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 68
• Tangents meet the radius at 90 degrees
• 112
• Opposite angles in a cyclic quad add to 180

D, E, F, G and H are points on a circle. Angle EGH = 67⁰

a) Find the size of angle EDH.

angle EDH = ​ ​ ​ ​ (a)   ​

b) Give a reason for your answer:

​ ​ ​ ​ (b)  

• 113
• Opposite angles in a cyclic quad add to 180
• The angle in a semi-circle is always 90 degrees.
• The angle in a semi-circle is always 90 degrees.
• Alternate segment theorem.
• Tangents which meet at an external point are equal
• 67
• 134
• Center angle = twice the circumference angle
• Tangents meet the radius at 90 degrees