### Angles On and Inside a Circle

When an angle is on a circle, the vertex is on the circumference of the circle. One type of angle *on* a circle is one formed by a tangent and a chord.

#### Investigation: The Measure of an Angle formed by a Tangent and a Chord

Tools Needed: pencil, paper, ruler, compass, protractor

- Draw with chord and tangent line with point of tangency .
- Draw in central angle . Then, using your protractor, find and .
- Find (the minor arc). How does the measure of this arc relate to ?

This investigation proves the Chord/Tangent Angle Theorem.

**Chord/Tangent Angle Theorem:** The measure of an angle formed by a chord and a tangent that intersect on the circle is half the measure of the intercepted arc.

From the Chord/Tangent Angle Theorem, we now know that there are two types of angles that are half the measure of the intercepted arc; an inscribed angle and an angle formed by a chord and a tangent. Therefore, ** any angle with its vertex on a circle will be half the measure of the intercepted arc**.

An angle is considered *inside* a circle when the vertex is somewhere inside the circle, but not on the center. All angles inside a circle are formed by two intersecting chords.

#### Investigation: Find the Measure of an Angle *inside* a Circle

Tools Needed: pencil, paper, compass, ruler, protractor, colored pencils (optional)

- Draw with chord and . Label the point of intersection .
- Draw central angles and . Use colored pencils, if desired.
- Using your protractor, find , and . What is and ?
- Find .
- What do you notice?

**Intersecting Chords Angle Theorem:** The measure of the angle formed by two chords that intersect *inside* a circle is the average of the measure of the intercepted arcs.

In the picture below:

#### Applying the Chord/Tangent Angle Theorem

1. Find

Use the Chord/Tangent Angle Theorem.

2. Find .

Use the Chord/Tangent Angle Theorem.

#### Measuring Angles

Find , and .

To find , it is in line with and . The three angles add up to . .

is an inscribed angle, so its measure is half of . From the Chord/Tangent Angle Theorem, .

.

To find , you can either use the Triangle Sum Theorem or the Chord/Tangent Angle Theorem. We will use the Triangle Sum Theorem. .

### Examples

Find .

Use the Intersecting Chords Angle Theorem and write an equation.

#### Example 1

The intercepted arcs for are and .

#### Example 2

Here, is one of the intercepted arcs for .

#### Example 3

is supplementary to the angle that the average of the given intercepted arcs. We will call this supplementary angle .

This means that

### Review

Find the value of the missing variable(s).

Solve for .

- Prove the Intersecting Chords Angle Theorem.

Given: Intersecting chords and .

Prove:

Fill in the blanks.

- If the vertex of an angle is _______________ a circle, then its measure is the average of the __________________ arcs.
- If the vertex of an angle is ________ a circle, then its measure is ______________ the intercepted arc.
- Can two tangent lines intersect inside a circle? Why or why not?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.7.