Circle Theorems

Spec reference: Geometry and measures - Properties and constructions
Key idea: Apply circle theorems to find missing angles.

---

Key circle vocabulary

• Radius - line from centre to circumference
• Diameter - line through the centre from one side to the other
• Chord - line joining two points on the circumference
• Tangent - line touching the circle at exactly one point
• Arc - part of the circumference
• Sector - pie-slice shaped region
• Segment - region between a chord and an arc

---

The circle theorems

Theorem	Statement
1	The angle at the centre is twice the angle at the circumference
2	Angles in a semicircle = 90° (angle in semicircle is a right angle)
3	Angles in the same segment are equal
4	Opposite angles in a cyclic quadrilateral add up to 180°
5	The tangent and radius meet at 90°
6	Tangents from an external point are equal in length
7	The perpendicular from the centre to a chord bisects it
8	Alternate segment theorem: angle between a tangent and chord = angle in the alternate segment

TIP

Always state the theorem you are using when giving reasons in exam answers.

---

Test Yourself

Question 1 of 5

Angle at centre is __ angle at circumference (same arc).