Circle Theorems Learning Outcomes Revise properties of

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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

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Circle Theorems
Circle Theorem 1
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
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Circle Theorems
Circle Theorem 2
Angles in the same segment are equal
D
C
A
B
Angle ACB Angle ADB
For angles subtended by the same arc are equal
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Circle Theorems
Circle Theorems
Example
Find angle CDE and CFE.
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Circle Theorems
Circle Theorems
Example
Find giving reasons i) ABO ii) AOB iii)
ADB
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Circle Theorems
Circle Theorems
Example
Find giving reasons i) BAC ii) ABD
38
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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
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Circle Theorems
Circle Theorems

• Draw Triangle ABC with B in 3 different positions
  on the circumference.

A

• Measure ABC for each of the 3 triangles.
• AB1C
• AB2C
• AB3C

• Complete the theorem

C
The angle in a semicircle is
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Circle Theorems
Circle Theorems
Find the unknown angles.
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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
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Circle Theorems
Circle Theorems
Tangent Kite
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.
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Circle Theorems
Circle Theorem 5
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
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Circle Theorems
Circle Theorem 5
Find all the missing angles in the diagram below,
also giving reasons.
i) BOA
A
C
x
O
40
B
ii) ACB
iii) ABX
iii) BAO
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Circle Theorems
Exam Question
(a) Explain why angle OTQ is 90
1
(b) Find the size of the angles (i)
TOQ (ii) OPT
1
1
(c) The angle RTQ is 57 Find the size of
the angle RUT
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
2
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Circle Theorems
Additional Notes
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Circle Theorems
Learning Outcomes At the end of the topic I will
be able to
Can Revise Do 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

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