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# Angles On and Inside a Circle

## Non-central angles whose vertices are on the circumference of a circle or formed by intersecting chords.

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Angles On and Inside a Circle

### 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

1. Draw A\begin{align*}\bigodot A\end{align*} with chord BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} and tangent line ED\begin{align*}\overleftrightarrow{ED}\end{align*} with point of tangency C\begin{align*}C\end{align*}.
2. Draw in central angle CAB\begin{align*}\angle CAB\end{align*}. Then, using your protractor, find mCAB\begin{align*}m \angle CAB\end{align*} and mBCE\begin{align*}m \angle BCE\end{align*}.
3. 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)

1. 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*}.
2. Draw central angles \begin{align*}\angle DAB\end{align*} and \begin{align*}\angle CAE\end{align*}. Use colored pencils, if desired.
3. 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*}?
4. Find \begin{align*}\frac{m \widehat{DB}+m \widehat{CE}}{2}\end{align*}.
5. 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).

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

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

1. 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.

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

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### Vocabulary Language: English Spanish

TermDefinition
central angle An angle formed by two radii and whose vertex is at the center of the circle.
chord A line segment whose endpoints are on a circle.
diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
inscribed angle An angle with its vertex on the circle and whose sides are chords.
intercepted arc The arc that is inside an inscribed angle and whose endpoints are on the angle.
point of tangency The point where the tangent line touches the circle.
Chord/Tangent Angle Theorem The Chord/Tangent Angle Theorem states that the measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.
Intersecting Chords Angle Theorem The Intersecting Chords Angle Theorem states that the measure of the angle formed by two chords that intersect inside a circle is the average of the measures of the intercepted arcs.