Inscribed Angle: Definition, Theorem & Formula

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  • 0:01 What Is an Inscribed Angle?
  • 0:44 What Is the Measure of…
  • 2:33 Multiple Inscribed…
  • 3:15 Lesson Summary
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Lesson Transcript
Instructor
Ellen Manchester
Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

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.

Inscribed Angle
Inscribed angle

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.

Intercepted Arc = 48 degrees, Inscribed angle = 24 degrees
48 degree arc measure

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.

Inscribed angle = 50 degrees
Inscribed angle = 50

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 degrees

2. Inscribed angle equals 20 degrees, what is the intercepted arc measure?

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Additional Activities

Verifying the Relationship Between Inscribed Angles and Intercepted Arcs

In this activity, we will use actual measurements to verify the relationship between the measure of an inscribed angle and the measure of the intercepted arc. The measure of an inscribed angle is one half the measure of its intercepted arc, or equivalently the measure of the intercepted arc is twice the measure of the inscribed angle.

Materials

  • Compass
  • Protractor
  • Paper and pencil

Directions

1) Make a point on the paper. Use the compass to draw a circle on the paper with the point you made as the center. The circle can be any size you like.

2) Create an inscribed angle on the circle : Place a point on the edge of the circle and, using the straight edge of the protractor, draw two lines from the point to other points on the circle. You can make your angle as wide or as thin as you want.

An example of what step 2 may look like

3) Use the protractor to measure your inscribed angle. Write down your measurement - be as precise as possible.

4) Use the protractor to measure the intercepted arc. To do this, center the protractor on the center of the circle and have 0 degrees on the protractor land on one end of the intercepted arc. Then, read the angle measurement on the protractor at the other end of the intercepted arc. Write down your measurement - be as precise as possible.

5) Multiply your measurement from step 3 by 2. Does it equal the measurement from step 4? (Slight differences are okay and a result of slight errors in measurement, but the result should be close!)

6) Repeat for other inscribed anhttps://study.com/academy/lesson/inscribed-angle-definition-theorem-formula.htmlgles or on circles of other sizes.

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