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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
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

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

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

If $$m\overparen{XZ} =40^{o}$$, what is the measure of inscribed angle $$Y$$?
$\red{ \text{m } \angle Y} = \frac 1 2 \overparen{XZ} \\ \red{ \text{m } \angle Y } = \frac 1 2 40 ^{\circ} \\ \red{ \text{m } \angle Y } = 20 ^{\circ}$
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 $$\text{ m } \overparen{YWZ} \ne 180^{\circ}.$$