Circle Theorems
Learning Outcomes



Revise properties of isosceles triangles, vertically opposite,
corresponding and alternate angles
Understand the terminology used – angle subtended by an arc or
chord
Use an investigative approach to find angles in a circle, to include:
• Angle in a semicircle
• Angle at centre and circumference
• Angles in the same segment
• Cyclic quadrilaterals
• Angle between tangent and radius and tangent kite
• Be able to prove and use the alternate segment theorem
Circle Theorem 1
Circle Theorems
The angle at the centre of a circle is double the size of the angle at the edge
D
O
A
B
Angle AOB = 2 x ADB
For angles subtended by the same arc, the angle at the centre is twice
the angle at the circumference
Circle Theorem 2
Circle Theorems
Angles in the same segment are equal
C
A
D
B
Angle ACB = Angle ADB
For angles subtended by the same arc are equal
Circle Theorems
Circle Theorems
Example:
Find angle CDE and CFE.
F
D
O
152°
C
E
Circle Theorems
Circle Theorems
Example:
Find giving reasons
i) ABO
ii) AOB
iii) ADB
D
O
40°
A
B
Circle Theorems
Circle Theorems
Example:
Find giving reasons
i) BAC
ii) ABD
D
38°
A
B
C
Circle Theorems
Circle Theorem 3
Opposite angles in a cyclic quadrilateral add up to 180°
Angle D + Angle B = 180°
Angle A + Angle C = 180°
A cyclic quadrilateral is a quadrilateral whose vertices all touch the
circumference of a circle. The opposite angles add up to 180°
Circle Theorems
Circle Theorems
1. Draw Triangle ABC with B
in 3 different positions on
the circumference.
A
2. Measure ABC for each of
the 3 triangles.
AB1C =
AB2C =
AB3C =
C
3. Complete the theorem :
The angle in a semicircle is
Circle Theorems
Circle Theorems
Find the unknown angles.
y
2x
72°
y
32°
x
3x
Circle Theorems
Circle Theorem 4
The angle between the tangent and the radius is 90°
The angle between a radius (or diameter) and a tangent is 90
This circle theorem gives rise to one ‘Tangent Kite’
Circle Theorems
Circle Theorems
‘Tangent Kite’
A
x
O
B
When 2 tangents are drawn from the point x a kite results. The
tangents are of equal length
BX = AX
Given OA = OB (radius)
OX is common the, the 2 triangles OAX and OBX are congruent.
Circle Theorem 5
Circle Theorems
Alternate Segment Theorem
Look out for a triangle with one of its vertices resting on the
point of contact of the tangent
Alternate segment
chord
tangent
The angle between a tangent and a chord is equal to the angle subtended
by the chord in the alternate segment
Circle Theorem 5
Circle Theorems
Find all the missing angles in the diagram below, also giving reasons.
i) BOA =
A
C
O
40°
x
B
ii) ACB =
iii) ABX =
iii) BAO =
Exam Question
Circle Theorems
P
(a) Explain why angle OTQ is 90 °
T
[1]
(b) Find the size of the angles
U
S
O
26°
R
In the diagram above, O is the centre
of the circle and PTQ is a tangent to
the circle at T. The angle POQ = 90°
and the angle SRT = 26°
Q
(i) TOQ
[1]
(ii) OPT
[1]
(c) The angle RTQ is 57°
Find the size of the angle RUT
[2]
Circle Theorems
Additional Notes
Circle Theorems
Learning Outcomes:
At the end of the topic I will be able to



Can
Do
Revise
Further
Revise properties of isosceles triangles, vertically
opposite, corresponding and alternate angles


Understand the terminology used – angle subtended by
an arc or chord


Use an investigative approach to find angles in a circle,
to include:
• Angle in a semicircle
• Angle at centre and circumference
• Angles in the same segment
• Cyclic quadrilaterals
• Angle between tangent and radius and tangent kite
• Be able to prove and use the alternate segment
theorem

