## Angles in circles

A **chord** of a circle is a straight line segment whose endpoints both lie on the circle (CD).

A **tangent** is a line which touches a circle at just one point. The radius to the point of tangency (A, B) is always **perpendicular** to the tangent line.
Conversely, the perpendicular to a radius through the same endpoint is a tangent line.

An **inscribed angle** in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle. Here, the circle with center O has the inscribed angle ∠ADB.
A **central angle** in a circle is formed by lines from two points on the circle’s circumference to the center of the circle. The vertex is located at the center of a circle. Here it is angle ∠AOB.
In a circle, the measure of a central angle is twice the measure of the inscribed angle with the same intercepted arc:

In a circle, two inscribed angles with the same intercepted arc are congruent (they have the same angle measure):

**Thales' theorem**: Any angle inscribed in a semi-circle is a right angle.

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

Angles of the same color have the same angle measure.

A **cyclic quadrilateral** has every vertex on a circle's circumference. A cyclic quadrilateral's opposite angles add to 180°: