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

#### Watch This

https://www.youtube.com/watch?v=aRGI4lafk8Y Brightstorm: Inscribed Angles

#### Guidance

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 examples.

**Example A**

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

**Solution:** 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.** In Example B, you will extend this proof.

**Example B**

Use the result from Example A to prove that .

**Solution:** Draw a diameter through points and .

From Example A, 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.**

**Example C**

Find .

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

**Concept Problem Revisited**

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.**

#### Vocabulary

A ** central angle** for a circle is an angle with its vertex at the center of the circle. The measure of a central angle is equal to the measure of its intercepted arc.

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.

An ** arc** is a portion of a circle. If an arc is less than half a circle it is called a

**. If an arc is more than half a circle it is called a**

*minor arc***.**

*major arc*
A ** chord** is a segment that connects two points on the circle. If a chord passes through the center of the circle then it is a

**.**

*diameter*#### Guided Practice

In #1-#2, you will use the circle below to prove that when two chords intersect inside a circle, the products of their segments are equal.

- Prove that .
*Hint: Look for congruent angles!* - Prove that .
- For the circle below, find .

**Answers:**

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

2. 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.**

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

#### Practice

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.

11.

12.

13.

14.

15. In the picture below, . Prove that .