Inscribed Angle

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.

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[inscribed angle] = (1/2)*m[intercepted arc]

m∠[brown] = (1/2) m[blue arc]

∠[brown]: Inscribed arc
[blue arc]: Intercepted arc

Example 1

Find the value of x. [measure of the inscribed angle]: x, [measure of the central angle]: 80.

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

Find the measure of arc ABC. [measure of angle APC]: 100.

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

Find the value of x. [measure of the inscribed angles]: 7x + 1, 3x + 29.

This brown angle is an inscribed angle
whose intercepted arc is the blue arc.

m∠[brown] = [7x + 1]

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

This brown angle is also an inscribed angle
whose intercepted arc is the same blue arc.

m∠[brown] = [3x + 29]

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

Both brown angles
have the same intercepted arc.

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

Move 3x to the left side.
And move +1 to the right side.

Then, 7x - 3x, 4x
is equal to,
+29 - 1, 28.

Divide both sides by 4.

Then x = 7.