In this lesson, we will look at finding angles in diagrams that involve tangents and circles.

Some of the theorems used are:

Tangent to Circle Theorem

Pythagorean Theorem

Two-Tangent Theorem

The following diagram shows the properties of the line segments and angles formed by the tangents from a point outside a circle. Scroll down the page for more examples and solutions on how to use the properties to solve for angles.

In the following diagram, *PA* and *PB* are tangents to the circle. Find the value of:

b) ∠AOB

c) ∠OBA

d) ∠ASB

e) the length of

* Solution*

Tangent to Circle Theorem

A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

Tangent to a Circle from a Point

Tangent segments to a circle from a point are congruent.

Tangent lines touches the circle at only one point

Point of Tangency is where the tangent line touches the circle

Examples:

Is AB a tangent line?

Find the missing angles.

Terms to know

A circle is a set of all points in a plane equidistant from a given point.

A radius is a segment whose endpoints are the center and a point on the circle.

A chord is a segment whose endpoints are on the circle.

A diameter is a chord that contains the center.

A secant is a line that a circle at two points.

A tangent is a line that intersects the circle at one point (point of tangency).

A common tangent is a line, ray or segment that is tangent to two coplanar circles.

Examples:

1. You are standing 14 feet from a water tower. The distance from you to the point of tangency on the tower is 28 feet. What is the radius of the water tower?

2. AB is a tangent to circle C at B. AD is a tangent to circle C at D. Find the value of x.

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