## The Lesson

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. 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. ## What Is a Circle?

A circle is a shape containing a set of points that are all the same distance from a given point, its center.

## Parts of a Circle

To understand the circle theorems, it is important to know the parts of a circle. ## Useful Definitions

Here are some useful definitions of some words used to explain the circle theorems.
• Perpendicular: means at right angles (90°) to.
• Bisector: to bisect means to divide into two equal parts.
• Subtended angle: the angle subtended by an arc is the angle between the lines drawn from the ends of the arc to the same point.
• Semicircle: half a circle.
• Cyclic quadrilateral: a four sided shape where each of the 4 corners lie on the circumference of a circle.
• Alternate segment: the alternate segment of a chord is formed by drawing lines from each end of the chord to a point on the circumference of the circle.