1. Circle Theorem Proofs Semi-Circle Centre Cyclic Quadrilateral
Same Segment
Alternate Segment
Tangents

2. Angle in a semi-circle is 90°
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Angle in a Semi-Circle
Angle in a semi-circle is 90°

3. => Angle in a semi-circle is 90°
PROOF:
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- First, draw a radius
=> 2 Isosceles Triangles
- Label an angle x°
= (180 – x)/2
= 90 - ½x
= 180 – x
= (180 - (180 – x))/2
= ½x
= 90°
=> Angle in a semi-circle is 90°
(or simply apply ‘angle at centre’)
Q.E.D

4. Angle at the centre is double the angle at the circumference
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Angle at the centre x°
Angle at the centre is double the angle at the circumference
2x°

5. => Angle at the centre double angle at the circumference
PROOF:
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- First, draw a radius
=> 2 Isosceles Triangles
= (180 – x)/2 x° y°
= (180 – y)/2
= 180 – ½x – ½y
= 360 – y - x
=> Angle at the centre double angle at the circumference
Q.E.D

6. Opposite Angles in a Cyclic Quadrilateral
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Opposite Angles in a Cyclic Quadrilateral x
Opposite angles in a cyclic quadrilateral add up to 180° 2y 2x y

7. => Opposite Angles in a Cyclic Quadrilateral add up to 180°
PROOF:
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First, draw in radii x
apply ‘angle at centre’ 2y
2x + 2y = 360º 2x
2(x + y) = 360º y
x + y = 180º
=> Opposite Angles in a Cyclic Quadrilateral add up to 180°
Q.E.D

8. Angles created by triangles are equal if they are in the same segment
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Angles in Same Segment
Angles created by triangles are equal if they are in the same segment

9. PROOF:
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- First, draw in radii

10. PROOF:
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- First, draw in radii
apply ‘angle at centre’ x 2x

11. => Angles in same segment are equal
PROOF:
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- First, draw in radii y
apply ‘angle at centre’ x
=> 2x = 2y 2y
x = y
=> Angles in same segment are equal
Q.E.D

12. Alternate Segment Theorem
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Alternate Segment Theorem
Angle between a tangent and a chord is equal to the angle at the circumference in the alternate segment
Chord
Chord
Tangent

13. PROOF: Start with two of the circle theorems
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Start with two of the circle theorems
Angle in a semi circle is 90°
Angle between a tangent and the radius is always 90°
and now combine them

14. => Angles in alternate segments are equal
PROOF:
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For cases when chord isn’t a diameter?
Label an angle x
90-x x
=> Angles in alternate segments are equal
Simply apply ‘same segment’ theorem
Q.E.D