Math Y10Y11 Y10 C3.6 IGCSE Maths Circle theorem Part 1

​The angle between a tangent to a circle and a chord at the point of contact is equal to the angle in the alternate segment.

​This theorem is called the alternate segment theorem

​tangent

​tangent

​The angle in the alternate segment, θ, is equal to the angle between the tangent and the chord.

​The chord divideds the circle into two segments. The segment that does not contain the angle between the tangent and the chord is the alternate segment.

​If the ends of the chord are joined to any point on the circle in the alternate segment, the angle between the lines is equal to the angle between the tangent and the chord.

Multiple Choice

What is the angle θ in the circle below?

• 45°

• 90°

• 180°

Multiple Choice

What is the angle θ in the circle below?

• 45°

• 90°

• 180°

Multiple Choice

What is the angle θ in the circle below?

• 45°

• 65°

• 115°

Multiple Choice

What is the angle Φ in the circle below?

• 53°

• 55°

• 72°

Multiple Choice

What is the angle θ in the circle below?

• 20°

• 52°

• 108°

Multiple Choice

 

Find the size of each lettered angle

• A= 65, b= 75, c= 40

• A= 75, b= 65, c=40

• A= 40, b= 75, c=65

Multiple Choice

Find the size of each lettered angle

• d=79, e=43, f=58

• d=58, e=43, f=79

• d=79, e=58, f=43

Multiple Choice

Find the size of each lettered angle

• g=41, h=76, i=76

• g=76, h=41, i=76

• g=41, h=76, i=41

Multiple Choice

Find the size of each lettered angle

• a = 75°

  b = 75°

  c = 75°

  d = 30°

• a = 75°

  b = 75°

  c = 75°

  d = 75°

• a = 75°

  b = 75°

  c = 75°

  d = 30°

Multiple Choice

Find the size of each lettered angle

• a = 47°

  b = 86°

  c = 86°

  d = 47°

• a = 86°

  b = 47°

  c = 86°

  d = 47°

• a = 86°

  b = 47°

  c = 47

  d = 86°

Multiple Choice

DAE is a tangent to a circle.

Write down the size of angle ABC. Give the reason for your answer

• 61 , alternate segments

• 61 , same segments

• 63, alternate segments

Multiple Choice

The straight line DCE is a tangent to the circle and it touches the circle at point C. The line AOC is the dimeter of the circle and the centre of the circle is the point labelled O.

Find the size of the following angles:

$$\angle$$  BCD

$$\angle$$  ABC = 90°

$$\angle$$  OCE

$$\angle$$  BCA

• $$\angle$$  BCD = 63°

  $$\angle$$  ABC = 90°

  $$\angle$$  OCE = 90°

  $$\angle$$  BCA = 27°

• $$\angle$$  BCD = 27°

  $$\angle$$  ABC = 90°

  $$\angle$$  OCE = 90°

  $$\angle$$  BCA =90°

• $$\angle$$  BCD = 27°

  $$\angle$$  ABC =90°

  $$\angle$$  OCE = 63°

  $$\angle$$  BCA = 90°

​

Circle Theorem 2: Angles at the centre and at the circumference

​

The angle at the centre is twice the angle at the circumference.

​

Circle Theorem 2: Angles at the centre and at the circumference

​

The angle at the centre is twice the angle at the circumference.

Multiple Choice

What is the angle at the centre?

• the angle at the centre is twice the angle at the circumference.

• the angle at the centre is same as the angle at the circumference.

• the angle at the centre is twice the angle at the segment.

• the angle at the centre is twice the angle at the radius

Multiple Choice

ABCD is an arrowhead where C is the centre of the circle and A, B, and D lie on the circumference. Calculate the size of angle BAD, $$\theta\degree$$  

• 75

• 150

• 30

• 180

Multiple Choice

Below is a circle with centre C. Points A, B, and D are on the circumference of the circle. AC = BC = radius. What is the name of this triangle ABC?

• Right angled

• Isosceles

• Scalene

Multiple Choice

Below is a circle with centre C. Points A, B, and D are on the circumference of the circle. AC = BC = radius. Calculate the angle BAC

• 90

• 46

• 180

Multiple Choice

Below is a circle with centre C. Points A, B, and D are on the circumference of the circle. Calculate the size of angle ACB.

• 46

• 92

• 88

• 23

Multiple Choice

Below is a circle with centre C. Points A, B, and D are on the circumference of the circle. Calculate the size of angle θ.

• 46

• 92

• 88

• 23

Multiple Choice

O is the centre of the circle. Write down the size of angle a. State the reason.

• 35, The angle at the centre is twice the angle at the circumference

• 140,The angle at the centre is twice the angle at the circumference

• 35, The angle at the circumference is twice the angle at the centre

• 140, The angle at the circumference is twice the angle at the centre

​Circle Theorem 3: Angles in the same segment

​

Angles in the same segment are equal.

Multiple Choice

A, B, C, D and E are points on a circle. AC is parallel to ED

Write down the size of angle x.

• 70

• 34

• 140

• 68

Multiple Choice

A, B, C and D are points on the circumference of a circle centre O. AC is a diameter. Angle DAC = 42°

Write down the value of x.

• 42

• 84

• 21

• 138

​Circle Theorem 4: Angles in a semicircle

​

The angle in a semicircle is 90 degrees.

Multiple Choice

A, B and C are points on the circumference of a circle with centre O. Work out the size of angle x.

• 45

• 35

• 55

• 90

​

Circle Theorem 5: Chord of a circle

​

The perpendicular from the centre of a circle to a chord bisects the chord (splits the chord into two equal parts).

​

Circle Theorem 6: Tangent of a circle

​

The angle between a tangent and radius is 90 degrees. Tangents which meet at the same point are equal in length.

​

Circle Theorem 7: Cyclic quadrilateral

​

The opposite angles in a cyclic quadrilateral total 180 degrees.

​This also highlights that angles inside a cyclic quadrilateral total 360°360°360°, which is the same property for any quadrilateral.

Multiple Choice

ABCD is a cyclic quadrilateral. Calculate the size of angle BCD

• 51

• 129

• 139

• 180

Multiple Choice

ABCD is a cyclic quadrilateral. Calculate the size of angle BAD :

• 75, The opposite angles in a cyclic quadrilateral total 180 degrees

• 42, The opposite angles in a cyclic quadrilateral total 180 degrees

• 75, The opposite angles in a cyclic quadrilateral total 360 degrees

• 42,The opposite angles in a cyclic quadrilateral total 360 degrees

Multiple Choice

ABCD is a cyclic quadrilateral with center O. Calculate the size of angle ABC.

• 81

• 55

• 145

• 116

Multiple Choice

ABCD is a cyclic quadrilateral inside a circle with centre O. BOC is the inscribed angle of the chord BC. What is the name of triangle BOC?

• Isosceles Triangle

• Equilateral Triangle

• Scalene Triangle

• Right Angled Triangle

Multiple Choice

ABCD is a cyclic quadrilateral inside a circle with centre O. BOC is the inscribed angle of the chord BC. Calculate the size of angle OBC.

• $$\frac{180-100}{2}$$  

• $$\frac{180-100}{3}$$  

• $$\frac{180+100}{2}$$  

• 180-100

Multiple Choice

ABCD is a cyclic quadrilateral inside a circle with centre O. BOC is the inscribed angle of the chord BC. Calculate the size of angle ADC.

• 88

• 92

• 102

• 80

Multiple Choice

Points A, B, C and D are on the circumference of the circle. CDE is a straight line. Work out the size of angle BCD. Give a reason for your answer.

• 108, Opposite angle of a cyclic quadrilateral add up to 180

• 125, Opposite angle of a cyclic quadrilateral add up to 360

• 180, Opposite angle of a cyclic quadrilateral is the same as 180

• 72, Opposite angle of a cyclic quadrilateral add up to 180