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Angles Inscribed in a Semicircle

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will demonstrate how to solve problems with angles inscribed in semi-circles. Inscribed angles and semi-circles are defined and examples are provided.

Inscribed Angles

Imagine it's a beautiful day and you would like to row your boat out on the lake. The lake happens to be a perfect circle, and you put in your boat at some point A of the lake rim. You row to the other side of the lake to some point C. Then you row back, but the wind pushes you away from your original entry point to another place, point B.

Your route describes an inscribed angle - an angle that is drawn inside a circle, where the vertex of the angle is a point on the circle.


Angle ACB is an inscribed angle and has vertex C on the circle.
Inscribed Angle


You decided to simply walk back from B to your starting point A. The path you would follow is the intercepted arc - the arc where the sides of the angle intersect the circle.


The red arc, AB, is the intercepted arc for the inscribed angle ACB
intercepted arc


As you walk, you realize that the measure of your arc is twice the measure of the angle you created at point B, the inscribed angle. Or, in other words, the measure of an inscribed angle is equal to half the measure of the intercepted arc.


inscribed angle measure


Example 1

Let's try this out with some real numbers. Find the measure of the arc AB.


example 1


The inscribed angle of ACB is 50 degrees. To find the measure of the intercepted arc, we simply multiply 50 by 2. The measure of the arc is 100 degrees.


answer 1


Example 2

Let's try another one. Find the measure of the angle ACB.


example 2


The intercepted arc measures 106 degrees. To find the measure of the inscribed angle, we divide 106 by 2 to get 53 degrees.


2 answer


Example 3

What if our angle isn't so perfect looking? Or our given numbers not so easy to plug in? Find the measure of the inscribed angle ACB.


example 3


In this case, we do not have the intercepted arc AB. Instead we are given the other two arcs that make up the circle. Since an entire circle is 360 degrees, we will add the two arcs together and subtract from 360 to get the arc AB.

89 + 153 = 242

360 - 242 = 118

Now that we know the intercepted arc is 118 degrees, we must divide it by 2 to get the angle ACB.


answer 3


Semi-Circles

A semi-circle is half of a circle.


semi circle


The endpoints of a semi-circle are the endpoints of a diameter (a segment that has endpoints on the circle and goes through the center of that circle).

If we inscribe an angle in a semi-circle, we can find the measure of the angle.


inscribed semi


A diameter is a line that goes through the center of the circle. A line measures 180 degrees. A diameter also cuts a circle in half, forming a semi-circle. If a circle is 360 degrees, then a semi-circle is 180 degrees.


line


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