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Inscribed Angle of a Circle and its intercepted arc

Definition, Formula and Practice

What is an Inscribed Angle ?


Is formed by 3 points that all lie on the circle's circumference.

Diagram 1
Inscribed Angle Example
The Formula

The measure of the inscribed angle is half of measure of the intercepted arc .

$ \text{m } \angle b = \frac 1 2 \overparen{AC} $

Explore this relationship in the interactive applet immediately below.

Interactive Inscribed Angle

$$ \angle D = \class{data-angle-0}{35.92} \\ \overparen{\rm BC} = \class{data-angle-1}{35.92} $$
Drag Points To Start Demonstration
Diagram 2
Picture of inscribed angles

Every single inscribed angle in diagram 2 has the exact same measure, since each inscribed angle intercepts the exact same arc, which is $$ \overparen {AZ} $$.

Practice Problems

Problem 1

Identify the inscribed angles and their intercepted arcs.

Inscribed Angle Diagram

$$ \overparen{XZ}\text{ is the intercepted arc} \\ \angle{Y}\text{ is the inscribed angle} \\ $$

Follow Up Question
If $$ m\overparen{XZ} =40^{o} $$, what is the measure of inscribed angle $$ Y $$?

$ \red{ \text{m } \angle Y} = \frac 1 2 \overparen{XZ} \\ \red{ \text{m } \angle Y } = \frac 1 2 40 ^{\circ} \\ \red{ \text{m } \angle Y } = 20 ^{\circ} $

Problem 2

Use your knowledge of the properties of inscribed angles and arcs to determine what is erroneous about the picture below.

Find the error in this inscribed angle problem

The error is that $$ 99^2 + 132^2 \ne 164^2 $$. Since the Pythagorean theorem does not hold, $$\angle X $$ is not a right angle and $$ \text{ m } \overparen{YWZ} \ne 180^{\circ}.$$

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