# Inscribed Angle

How to find the inscribed angle of a circle: definition, formula, examples, and their solutions.

## Definition

An inscribed angle is an angle

whose vertex is on the circle

and whose sides are the chords of the circle.

Chord of a circle

## Formula

m∠[brown] = (1/2) m[blue arc]

∠[brown]: Inscribed arc

[blue arc]: Intercepted arc

## Example 1

m∠[blue] = 80

So m[blue arc] = 80.

Measure of an arc

m[blue arc] = 80

So *x* = (1/2)⋅80.

(1/2)⋅80 = 40

So *x* = 40.

## Example 2

m∠[brown] = 100

So 100 = (1/2) m[arc *ABC*].

Switch both sides.

Multiply 2 on both sides.

Then m[arc *ABC*] = 200.

## Example 3

This brown angle is an inscribed angle

whose intercepted arc is the blue arc.

m∠[brown] = [7*x* + 1]

So [7*x* + 1] = (1/2) m[blue arc].

This brown angle is also an inscribed angle

whose intercepted arc is the same blue arc.

m∠[brown] = [3*x* + 29]

So (1/2) m[blue arc] = [3*x* + 29].

Both brown angles

have the same intercepted arc.

So [7*x* + 1] = [3*x* + 29].

Move 3*x* to the left side.

And move +1 to the right side.

Then, 7*x* - 3*x*, 4*x*

is equal to,

+29 - 1, 28.

Divide both sides by 4.

Then *x* = 7.