### 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 \begin{align*}\bigodot A\end{align*}
⨀A with chord \begin{align*}\overline{BC}\end{align*}BC¯¯¯¯¯¯¯¯ and tangent line \begin{align*}\overleftrightarrow{ED}\end{align*}ED←→ with point of tangency \begin{align*}C\end{align*}C . - Draw in central angle \begin{align*}\angle CAB\end{align*}
∠CAB . Then, using your protractor, find \begin{align*}m \angle CAB\end{align*}m∠CAB and \begin{align*}m \angle BCE\end{align*}m∠BCE . - 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*}

#### Applying the Chord/Tangent Angle Theorem

1. 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*}

2. 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*}

#### Measuring Angles

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*}.

### Examples

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

Use the Intersecting Chords Angle Theorem and write an equation.

#### Example 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*}

#### Example 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*}

#### Example 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*}

### Review

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?

### Review (Answers)

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