# 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].

m[blue arc] = 80

x = (1/2)⋅80

## Example 2

m∠APC = 100

100 = (1/2)⋅m[arc ABC]

## Example 3

7x + 1 = m[blue arc]
3x + 29 = m[blue arc]

So 7x + 1 = 3x + 29.