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°: