Geometry (B)

Angles of Chords, Secants, and Tangents

Learning Objective

• find the measures of angles formed by chords, secants, and tangents

Measure of Tangent-Chord Angle

Theorem The measure of an angle formed by a chord and a tangent that intersect on the circle equals half the measure of the intercepted arc.

In other words:

and

Proof

Draw the radii of the circle to points and .

is isosceles, therefore

We also know that, because is tangent to the circle.

We obtain

Since is a central angle that corresponds to then, .

This completes the proof.

Example 1

Find the values of and .

First we find angle

Using the Measure of the Tangent Chord Theorem we conclude that:

and

Therefore,

Angles Inside a Circle

Theorem The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measure of their intercepted arcs. In other words, the measure of the angle is the average (mean) of the measures of the intercepted arcs.

In this figure,

Proof

Draw a segment to connect points and .

Example 2

Find .

Angles Outside a Circle

Theorem The measure of an angle formed by two secants drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.

In other words:

This theorem also applies for an angle formed by two tangents to the circle drawn from a point outside the circle and for an angle formed by a tangent and a secant drawn from a point outside the circle.

Proof

Draw a line to connect points and .

Example 3

Find the measure of angle .

Lesson Summary

In this section we learned about finding the measure of angles formed by chords, secants, and tangents. We looked at the relationship between the arc measure and the angles formed by chords, secants, and tangents.

The questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

1. Find the value of the variable.
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2. Find the measure of the following angles:

   

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3. Find the measure of the following angles:

   

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4. Four points on a circle divide it into four arcs, whose sizes are , and , in consecutive order. The four points determine two intersecting chords. Find the sizes of the angles formed by the intersecting chords.

Review Answers

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