What if you were given a circle with either a chord and a tangent or two chords that meet at a common point? How could you use the measure of the arc(s) formed by those circle parts to find the measure of the angles they make on or inside the circle? After completing this Concept, you'll be able to apply the Chord/Tangent Angle Theorem and the Intersecting Chords Angle Theorem to solve problems like this one.

### Watch This

CK-12 Foundation: Chapter9AnglesOnandInsideaCircleA

Learn more about chords and tangents by watching the second part of the video at this link.

Follow this link to watch a video about secants.

### Guidance

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
⨀A with chordBC¯¯¯¯¯¯¯¯ and tangent lineED←→ with point of tangencyC . - Draw in central angle
∠CAB . Then, using your protractor, findm∠CAB andm∠BCE . - Find
mBCˆ (the minor arc). How does the measure of this arc relate tom∠BCE ?

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
⨀A with chordBC¯¯¯¯¯¯¯¯ andDE¯¯¯¯¯¯¯¯ . Label the point of intersectionP . - Draw central angles
∠DAB and∠CAE . Use colored pencils, if desired. - Using your protractor, find
m∠DPB,m∠DAB , andm∠CAE . What ismDBˆ andmCEˆ ? - Find
mDBˆ+mCEˆ2 . - 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:

#### Example A

Find

Use the Chord/Tangent Angle Theorem.

#### Example B

Find

Use the Chord/Tangent Angle Theorem.

#### Example C

Find

To find

To find

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter9AnglesOnandInsideaCircleB

### Vocabulary

A ** circle** is the set of all points that are the same distance away from a specific point, called the

**. A**

*center***is the distance from the center to the circle. A**

*radius***is a line segment whose endpoints are on a circle. A**

*chord***is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A**

*diameter***is the angle formed by two radii and whose vertex is at the center of the circle. An**

*central angle***is an angle with its vertex on the circle and whose sides are chords. The**

*inscribed angle***is the arc that is inside the inscribed angle and whose endpoints are on the angle. A**

*intercepted arc***is a line that intersects a circle in exactly one point. The**

*tangent***is the point where the tangent line touches the circle.**

*point of tangency*### Guided Practice

Find

1.

2.

3.

**Answers:**

Use the Intersecting Chords Angle Theorem and write an equation.

1. The intercepted arcs for

2. Here,

3.

### Interactive Practice

### Practice

Find the value of the missing variable(s).

- \begin{align*}y \ne 60^\circ\end{align*}

Solve for \begin{align*}x\end{align*}.

- Prove the Intersecting Chords Angle Theorem.

Given: Intersecting chords \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{BD}\end{align*}.

Prove: \begin{align*}m\angle a=\frac{1}{2} \left (m\widehat{DC}+m\widehat{AB}\right )\end{align*}

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?