Circles are all around us in our world. Inscribed angles are angles that sit inside a circle with the vertex on the circumference of the circle.

Inscribed angles have a special relationship with the intercepted arc.

## What Is an Inscribed Angle?

An **inscribed angle** is an angle whose vertex sits on the circumference of a circle. The vertex is the common endpoint of the two sides of the angle. The two sides are chords of the circle. A chord is a line segment whose endpoints also sit on the circumference of a circle.

One endpoint is the vertex while the other endpoint sits across the circle. The arc formed by the inscribed angle is called the **intercepted arc**. This arc is part of the circumference of the circle that is between the two chords of the angle, or intercepted by the chords.

The intercepted arc and the inscribed angle have a special relationship.

## What Is the Measure of the Inscribed Angle?

An inscribed angle is half the measure of its intercepted arc.

If you know the inscribed angle measure, you can figure out the intercepted arc measure. If you know the intercepted arc measure you can figure out the inscribed angle measure. Let’s try a few.In this example, we have an intercepted arc measure of 48 degrees. If the inscribed angle is half of its intercepted arc, half of 48 equals 24.

So, the inscribed angle equals 24 degrees. 48 * 1/2 is the same as 48 / 2. Both equal 24.

If you have an inscribed angle of 50 degrees, what is the intercepted arc measure? Remember the inscribed angle is half the measure of its intercepted arc, so 50 * 2 = 100 degrees.

When the inscribed angle is 50 degrees, the intercepted arc measure is 100 degrees.

Here are a few more for you to try on your own.

Remember, the inscribed angle is half the intercepted arc measure.1. Inscribed angle equals 36 degrees, what is the intercepted arc measure?The answer is: 36 * 2 = 72 degrees2. Inscribed angle equals 20 degrees, what is the intercepted arc measure?The answer is 20 * 2 = 40 degrees3. Intercepted arc measure equals 120 degrees, what is the inscribed angle measure?The answer is: 120 / 2 = 60 degrees4. Intercepted arc measure equals 150 degrees, what is the inscribed angle measure?The answer is: 150 / 2 = 75 degrees

## Multiple Inscribed Angles ; the Intercepted Arc

Take a look at this circle.

What do you notice about the intercepted arc?

Notice you have three different inscribed angles, but all three have the same intercepted arc. If the sides of the angle encompass the same arc, then the three inscribed angles have the same measure.In this example, all three inscribed angles have a measure of 37.

4 degrees. The intercepted arc equals twice this value, so do you know it? Yes, 37.4 * 2 = 74.8 degrees. So, three different inscribed angles of 37.4 degrees all have the same intercepted arc measure of 74.

8 degrees.

## Lesson Summary

The **inscribed angle** is an angle whose vertex sits on the circumference of a circle and whose sides are chords of the circle. The arc formed by the inscribed angle is called the **intercepted arc**. To find the inscribed angle, cut the intercepted arc in half. To find the intercepted arc, multiply the inscribed angle by two.