Property of the Angle Formed by a Tangent and a Chord of a Circle

The angle formed by a tangent to a circle and a chord passing through the point of tangency is equal in measure to the inscribed angle subtended by the arc inside that angle.

n the following diagram, line is the tangent to the circle at point (the point of tangency), and

Example 1. In the figure below, is a tangent of the circle and is the point of contact. Find the size of .

 Solution

The size of the angle between a tangent of a circle and the chord passing the point of contact is same as the size of the inscribed angle which is made from the chord.

Example 2. In the figure below, is a tangent of the circle and is the point of contact. Find the size of when and .

 Solution

Since ,
For ,