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

**. You can also say that**

*inscribed in*

*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.

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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.