# Circle Theorems

## What Are the Circle Theorems?

Circles have properties relating to angles and lines.

There are several circle theorems that apply to all circles.

### A Tangent and a Radius Meet at 90°

The tangent makes 90° with the radius which it meets at the point at which it touches.

### Two Radii Form an Isosceles Triangle

Two radii form the two equal sides of an isosceles triangle.

**Note:** Radii is the plural of radius.

### The Perpendicular Bisector of a Chord Passes Through the Center of the Circle

If a line cuts through a chord of the circle, such that it crosses it at 90° and cuts it in half, that line passes through the centre of the circle.

### The Angle at the Center of a Circle Is Twice the Angle at the Circumference

The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference.

### The Angle in a Semicircle Is 90°

A triangle drawn from two ends of a diameter makes 90° at the circumference.

### Angles in the Same Segment Are Equal

All triangles drawn from a chord make the same angle at the circumference.

### Opposite Angles in a Cyclic Quadrilateral Add Up to 180°

A cyclic quadrilateral is a 4 sided shape where each corner touches the circle. Both pairs of opposite angles add up to 180°.

### Tangents from the Same Point Are the Same Length

Two tangents drawn from the same point outside of the circle are the same length and form two congruent right triangles.

### The Angle Between a Tangent and a Chord is Equal to the Angle in the Alternate Segment

The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.