# Circle Theorems

These lessons, with videos, examples and step-by-step solutions help GCSE/IGCSE Maths students learn the circle theorems.

### Circle Theorems Summary

• Angle subtended at the centre of a circle is twice the angle at the circumference.
• The angle between a radius and a tangent is 90 degrees.
• The angle at the centre is twice the angle at the circumference.
• Angles in the same segment are equal.
• The angle in a semi-circle is always 90 degrees.
• The opposite angles in a cyclic quadrilateral always add up to 180 degrees.
• The angle between a circle and a tangent is equal to the angle in the alternate segment.
• The lengths from where two tangents touch a circle to where they meet each other are equal.

The following diagram shows some circle theorems: angle in a semicircle, angle between tangent and radius of a circle, angle at the centre of a circle is twice the angle at the circumference, angles in the same segment are equal, angles in opposite segments are supplementary; cyclic quadrilaterals and alternate segment theorem. Scroll down the page for more examples and solutions of circle theorems. Circle Theorem Basic definitions
Chord, segment, sector, tangent, cyclic quadrilateral.
Theorem: Angle subtended at the centre of a circle is twice the angle at the circumference.

Circle Theorems part 1 of 2
The angle between a radius and a tangent is 90 degrees.
The angle at the centre is twice the angle at the circumference.
Angles in the same segment are equal.
The angle in a semi-circle is always 90 degrees.
The opposite angles in a cyclic quadrilateral always add up to 180 degrees.

Circle Theorems part 2 of 2
The angle between a circle and a tangent is equal to the angle in the alternate segment.
The lengths from where two tangents touch a circle to where they meet each other are equal.

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