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An angle can be constructed by the lines and line segments that intersect a circle. In this lesson, the angles related to circles and their properties will be explored.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Explore

Investigating Inscribed Angles of a Circle

Consider a circle with a radius of $4$ units. An angle whose sides are two chords of the circle is formed as shown. Move the points $B$ and $C$ along the circle so that the angle is a right angle.

Think about the following questions.

When the measure of the angle is $9 0^{\circ},$ what can the chord joining $B$ and $C$ be called?
Does the arc that the angle intercepts have a special name? If yes, what is it?

Explore

Inscribed and Central Angles of a Circle

On the circle above, construct an angle with a vertex at the center of the circle. The angle being constructed should also cut off the same arc. In other words, construct the corresponding central angle.

Recall that the measure of an arc is the measure of the central angle that includes the arc. Now, move the points and make conjectures that explain the relationship between the measures of angles and arcs.

Discussion

Inscribed Angle of a Circle

In the circles shown previously, the angle formed by two chords $∠ B A C$ is called an inscribed angle.

Discussion

Inscribed Angle Theorem

Observe the measures of the angle and the arc intercepted by the angle. Start by moving $B$ so that $B,$ $O,$ and $A$ are collinear. Then, move it once again so that $B,$ $O,$ and $C$ are collinear.

As can be seen, when $B,$ $O,$ and $C$ are collinear, $\overline{B C}$ becomes the diameter of the circle, and the angle cuts off the semicircle. Furthermore, the measure of $B C$ is twice the measure of an inscribed angle that intercepts it. This statement can be restated as a theorem.

Example

Finding the Inscribed Angle of a Circle

In the diagram, the vertex of $∠ U V T$ is on the circle $O,$ and the sides of the angle are chords of the circle. Given the measure of $U T$ , find the measure of the angle.

Write the answer without the degree symbol.

Hint

The Inscribed Angle Theorem can be used to find the measure of the angle.

Solution

The angle shown in the diagram fits the definition of an inscribed angle. For this reason, the measure of the angle can be found using the Inscribed Angle Theorem. The theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

The arc intercepted by

∠ U V T

measures

4 0^{\circ} .

m ∠ U V T = \frac{1}{2} m U T

$m U T = 4 0^{\circ}$

m ∠ U V T = \frac{1}{2} (4 0^{\circ})

Multiply

m ∠ U V T = 2 0^{\circ}

Therefore,

∠ U V T

measures

2 0^{\circ} .

Pop Quiz

Practice Finding Inscribed Angles

Find the measure of the inscribed angle in the circle.

Inscribed Angle

Example

Finding the Central Angle of a Circle

Similarly, given the measure of an inscribed angle, the measure of its corresponding central angle can be found using the Inscribed Angle Theorem. This can be done because the measure of the central angle is the same as the measure of the arc that the central angle cuts off.

In the circle, $∠ K L M$ measures $6 7^{\circ} .$

Find the measure of the corresponding central angle.

Hint

Start by drawing the corresponding central angle.

Solution

Recall that a central angle is an angle whose vertex lies at the center of the circle. Additionally, the inscribed angle and its corresponding central angle intercept the same arc $K M$ for this example. Therefore, the corresponding central angle is $∠ K O M .$

The arc

K M

is included in

∠ K O M .

Therefore, the measure of

∠ K O M

is equal to the measure of

K M .

Using the Inscribed Angle Theorem, the measure of

K M

can be found.

m ∠ K L M = \frac{1}{2} m K M

$m ∠ K L M = 6 7^{\circ}$

6 7^{\circ} = \frac{1}{2} m K M

Solve for

m K M

$LHS \cdot 2 = RHS \cdot 2$

134 = m K M

Rearrange equation

m K M = 13 4^{\circ}

Therefore,

∠ K O M

also measures

13 4^{\circ} .

Pop Quiz

Practice Finding Central Angles

Given the measure of an inscribed angle, find the measure of its corresponding central angle.

Inscribed Angle

Discussion

Inscribed Angles of a Circle Theorem

Up to now, the relationship between inscribed angles and their corresponding central angles has been discussed. Now the relationship between two inscribed angles that intercept the same arc will be investigated.

As can be observed, the angles are congruent, so long as they intercept the same arc.

Discussion

Circumscribed Angle Theorem

A circumscribed angle is supplementary to the central angle it cuts off.

Pop Quiz

Practice Using the Circumscribed Angle Theorem

Find the measure of the central angle.

Inscribed Angle

Example

Using Circle Theorems to Solve Problems

The following example involving circumscribed angles and inscribed angles could require the use of the previously learned theorems.

Two tangents from $P$ to $⊙ M$ are drawn. The measure of $∠ K P L$ is $4 0^{\circ} .$

Find the measure of the inscribed angle that intercepts the same arc as $∠ K P L ?$

Hint

Use the Circumscribed Angle Theorem and Inscribed Angle Theorem.

Solution

The inscribed angle that intercepts

K L

is

∠ K N L .

By the Inscribed Angle Theorem,

m ∠ K N L

is half

m K L .

m ∠ K N L = \frac{1}{2} m K L

Recall that the measure of an arc is the same as the measure of its corresponding central angle. Therefore,

m ∠ K M L

is equal to

m K L .

Substitute

m ∠ K M L

into the above equation.

m ∠ K N L m ∠ K N L = \frac{1}{2} m K L = \frac{1}{2} m ∠ K M L

Now, by the Circumscribed Angle Theorem,

m ∠ K M L

is

18 0^{\circ}

minus

m ∠ K P L .

m ∠ K N L = \frac{1}{2} (18 0^{\circ} - m ∠ K P L)

It is given that

m ∠ K P L = 4 0^{\circ} .

Substituting that measure into the equation will give the measure of the inscribed angle.

m ∠ K N L = \frac{1}{2} (18 0^{\circ} - m ∠ K P L)

$m ∠ K P L = 4 0^{\circ}$

m ∠ K N L = \frac{1}{2} (18 0^{\circ} - 4 0^{\circ})

Evaluate right-hand side

Subtract term

m ∠ K N L = \frac{1}{2} (14 0^{\circ})

Multiply

m ∠ K N L = 7 0^{\circ}

Closure

Summarizing Angle Theorems Related to Circles

This lesson defined three angles related to circles as well as the relationships between these angles. The diagram below shows the definitions and the main theorems of this lesson.

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