# Inscribed Angle

How to solve inscribed angle problems: definition, property, proof, 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.

## Property

m∠(inscribed angle, red) = (1/2)⋅m[intercepted arc, blue].

## Proof

Case 1: The center of the circle

is on the side of an inscribed angle.

Draw a radius to make an isosceles triangle.

Then the red angles are congruent.

Then m∠(red) + m∠(red) = m∠(blue).

Exterior angle of a triangle

So 2m∠(red) = m∠(blue).

m∠(blue) = m[blue arc].

Measure of an arc

So 2m∠(red) = m∠(blue) = m[blue arc].

Divide both sides by 2.

Then m∠(red) = (1/2)⋅m[blue arc].

Case 2: The center of the circle

is in the interior of an inscribed angle.

Draw a dashed diameter.

(It passes through the center of the circle.)

Set the measures of the divided arcs as *x* and *y*.

Then *x* + *y* = m[blue arc].

By Case 1,

the measures of the divided inscribed angles are

(1/2)*x* and (1/2)*y*.

Then m∠(red) = (1/2)*x* + (1/2)*y*.

*x* + *y* = m[blue arc]

So m∠(red) = (1/2)(*x* + *y*)

= (1/2)⋅m[blue arc].

Case 3: The center of the circle

is on the side of an inscribed angle.

Draw a dashed diameter.

(It passes through the center of the circle.)

Set the measure of the extended arc as *x*.

And set the measure of the exterior arc as *y*.

Then *x* - *y* = m[blue arc].

By Case 1,

the measure of the extended inscribed angle is (1/2)*x*,

and the measure of the excluded inscribed angle is (1/2)*y*.

Then m∠(red) = (1/2)*x* - (1/2)*y*.

*x* - *y* = m[blue arc]

So m∠(red) = (1/2)(*x* - *y*)

= (1/2)⋅m[blue arc].

## Example 1

m[blue arc] = 80*x* = (1/2)⋅80

## Example 2

m∠*APC* = 100

100 = (1/2)⋅m[arc *ABC*]

## Example 3

7*x* + 1 = m[blue arc]

3*x* + 29 = m[blue arc]

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