How to Prove the Inscribed Angle Theorem
An angle which sits on the boundary of a circle is called an inscribed angle.
An angle which sits in the center of a circle is called a central angle.
Formula
Inscribed Angle Theorem
If the inscribed angle spans the same circular sector as the central angle , then the central angle is twice the size of the inscribed angle:
Think About This
Proof of the Inscribed Angle Theorem
Look closely at the figure below!
From the figure you get the following information:
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, as they are all radii of the circle.
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Thus, and both have two equally sized legs.
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This means that ,
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and .
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Thus, .
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You also know that .
When combining all this information, you get:
Example 1
Find all the angles in the following triangles:
When you know the inscribed angle, you can find the central angle, because it is always exactly two times larger. Thus
Triangle has two equal sides as —they are both radiuses of the circle. Thus
Triangle has an angle ; you need to find angles and . You already know that angle , so you get
The final angle is then
You have now found all the angles in both triangles.
