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 \begin{align*}\bigodot A\end{align*} with chord \begin{align*}\overline{BC}\end{align*} and tangent line \begin{align*}\overleftrightarrow{ED}\end{align*} with point of tangency \begin{align*}C\end{align*}.
- Draw in central angle \begin{align*}\angle CAB\end{align*}. Then, using your protractor, find \begin{align*}m \angle CAB\end{align*} and \begin{align*}m \angle BCE\end{align*}.
- Find \begin{align*}m \widehat{BC}\end{align*} (the minor arc). How does the measure of this arc relate to \begin{align*}m \angle BCE\end{align*}?

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 \begin{align*}\bigodot A\end{align*} with chord \begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{DE}\end{align*}. Label the point of intersection \begin{align*}P\end{align*}.
- Draw central angles \begin{align*}\angle DAB\end{align*} and \begin{align*}\angle CAE\end{align*}. Use colored pencils, if desired.
- Using your protractor, find \begin{align*}m \angle DPB, m \angle DAB\end{align*}, and \begin{align*}m \angle CAE\end{align*}. What is \begin{align*}m \widehat{DB}\end{align*} and \begin{align*}m \widehat{CE}\end{align*}?
- Find \begin{align*}\frac{m \widehat{DB}+m \widehat{CE}}{2}\end{align*}.
- 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:

\begin{align*}m \angle SVR &= \frac{1}{2} \left(m \widehat{SR}+m \widehat{TQ}\right)=\frac{m \widehat{SR}+m\widehat{TQ}}{2}=m \angle TVQ\\ m \angle SVT &= \frac{1}{2} \left( m \widehat{ST}+m \widehat{RQ}\right)=\frac{m \widehat{ST}+m \widehat{RQ}}{2}=m \angle RVQ\end{align*}

#### Example A

Find \begin{align*}m \widehat{AEB}\end{align*}

Use the Chord/Tangent Angle Theorem.

\begin{align*}m \widehat{AEB} =2 \cdot m \angle DAB = 2 \cdot 133^\circ=266^\circ\end{align*}

#### Example B

Find \begin{align*}m \angle BAD\end{align*}.

Use the Chord/Tangent Angle Theorem.

\begin{align*}m \angle BAD=\frac{1}{2} m \widehat{AB} =\frac{1}{2} \cdot 124^\circ=62^\circ\end{align*}

#### Example C

Find \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*}.

To find \begin{align*}a\end{align*}, it is in line with \begin{align*}50^\circ\end{align*} and \begin{align*}45^\circ\end{align*}. The three angles add up to \begin{align*}180^\circ\end{align*}. \begin{align*}50^\circ + 45^\circ + m \angle a = 180^\circ, m \angle a = 85^\circ\end{align*}.

\begin{align*}b\end{align*} is an inscribed angle, so its measure is half of \begin{align*}m \widehat{AC}\end{align*}. From the Chord/Tangent Angle Theorem, \begin{align*}m \widehat{AC} =2 \cdot m \angle EAC=2 \cdot 45^\circ=90^\circ\end{align*}.

\begin{align*}m \angle b=\frac{1}{2} \cdot m \widehat{AC} =\frac{1}{2} \cdot 90^\circ=45^\circ\end{align*}.

To find \begin{align*}c\end{align*}, you can either use the Triangle Sum Theorem or the Chord/Tangent Angle Theorem. We will use the Triangle Sum Theorem. \begin{align*}85^\circ + 45^\circ + m \angle c = 180^\circ, m \angle c = 50^\circ\end{align*}.

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

1.

2.

3.

**Answers:**

Use the Intersecting Chords Angle Theorem and write an equation.

1. The intercepted arcs for \begin{align*}x\end{align*} are \begin{align*}129^\circ\end{align*} and \begin{align*}71^\circ\end{align*}.

\begin{align*}x=\frac{129^\circ+71^\circ}{2}=\frac{200^\circ}{2}=100^\circ\end{align*}

2. Here, \begin{align*}x\end{align*} is one of the intercepted arcs for \begin{align*}40^\circ\end{align*}.

\begin{align*}40^\circ &= \frac{52^\circ+x}{2}\\ 80^\circ &= 52^\circ+x\\ 38^\circ &= x\end{align*}

3. \begin{align*}x\end{align*} is supplementary to the angle that the average of the given intercepted arcs. We will call this supplementary angle \begin{align*}y\end{align*}.

\begin{align*}y=\frac{19^\circ+107^\circ}{2}=\frac{126^\circ}{2}=63^\circ\end{align*} This means that \begin{align*}x=117^\circ; 180^\circ-63^\circ\end{align*}

### 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?