Circle Theorems

Understanding Circle Theorems

Circle theorems are rules that help us understand the properties and relationships within a circle. These theorems are essential for solving geometric problems involving circles.

Key Circle Theorems

• Angle at the Centre Theorem: The angle subtended at the centre of a circle is twice the angle subtended at the circumference by the same arc.
• Angles in the Same Segment: Angles in the same segment of a circle are equal.
• Angle in a Semicircle: An angle subtended by a diameter on the circumference is a right angle (90°).
• Cyclic Quadrilateral: The opposite angles of a cyclic quadrilateral (a four-sided figure with all vertices on the circle) sum up to 180°.
• Tangent-Secant Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Worked Example

Example

If a cyclic quadrilateral ABCD is inscribed in a circle, and $\angle ABC = 110°$, find $\angle ADC$.

• According to the Cyclic Quadrilateral Theorem, opposite angles sum up to 180°.
• Therefore, $\angle ADC = 180° - \angle ABC = 180° - 110° = 70°$