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# Inscribed Angles in Circles

## Angle with its vertex on a circle and sides that contain chords.

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Inscribed Angles

Point  is the center of the circle below. What can you say about ?

### Inscribed Angles

An inscribed angle is an angle with its vertex on the circle. The sides of an inscribed angle will be chords of the circle. Below,  is an inscribed angle.

Inscribed angles are inscribed in arcs. You can say that  is inscribed in . You can also say that  intercepts .

The measure of an inscribed angle is always half the measure of the arc it intercepts. You will prove and then use this theorem in the problems below.

#### Proving the Inscribed Angle Theorem

1. Consider the circle below with center at point . Prove that .

and  are both radii of the circle and are therefore congruent. This means that  is isosceles. The base angles of isosceles triangles are congruent so . is an exterior angle of , so its measure must be the sum of the measures of the remote interior angles. Therefore, . By substitution, . This means  and . A central angle has the same measure as its intercepted arc, so . Therefore by substitution, .

This proves that when an inscribed angle passes through the center of a circle, its measure is half the measure of the arc it intercepts.

2. Use the result from the previous problem to prove that .

Draw a diameter through points  and .

From #1, you know that .

You also know that .

Because , by substitution . This means . , so .

Because .

This proves in general that the measure of an inscribed angle is half the measure of its intercepted arc.

Now let's find the measure of an angle.

Find .

Notice that both  and  intercept . This means that their measures are both half the measure of , so their measures must be equal. .

### Examples

#### Example 1

Earlier, you were asked what can you say about

Point  is the center of the circle below. What can you say about ?

If point  is the center of the circle, then  is a diameter and it divides the circle into two equal halves. This means that is an inscribed angle that intercepts , so its measure must be half the measure of . Therefore,  and  is a right triangle.

In general, if a triangle is inscribed in a semicircle then it is a right triangle.

In #2-#3, you will use the circle below to prove that when two chords intersect inside a circle, the products of their segments are equal.

#### Example 2

Prove that . Hint: Look for congruent angles!

because both are inscribed angles that intercept the same arc because they are vertical angles. Therefore, by .

#### Example 3

Prove that .

Because , its corresponding sides are proportional. This means that . Multiply both sides of the equation by and you have . This proves that in general, when two chords intersect inside a circle, the products of their segments are equal.

#### Example 4

For the circle below, find .

Based on the result of #2, you know that . This means that .

### Review

1. How are central angles and inscribed angles related?

In the picture below, . Use the picture below for #2-#6.

2. Find .

3. Find .

4. Find .

5. Find .

6. What type of triangle is ?

Solve for  in each circle. If  is an angle, find the measure of the angle.

7.

8.

9.

10.

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15. In the picture below, . Prove that .

### Review (Answers)

To see the Review answers, open this  PDF file and look for section 8.4.

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### Vocabulary Language: English

Arc

An arc is a section of the circumference of a circle.

Intercepts

The intercepts of a curve are the locations where the curve intersects the $x$ and $y$ axes. An $x$ intercept is a point at which the curve intersects the $x$-axis. A $y$ intercept is a point at which the curve intersects the $y$-axis.

Inscribed Angle Theorem

The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

Semicircle Theorem

The Semicircle Theorem states that any time a right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter and the diameter is the hypotenuse.

Inscribed Angle

An inscribed angle is an angle with its vertex on the circle. The measure of an inscribed angle is half the measure of its intercepted arc.

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