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[inscribed angle] = (1/2)*m[intercepted arc]](https://ximpledu.com/en-us/inscribed-angle/inscribed-angle-02-02.png "Inscribed Angle: Formula")
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.](https://ximpledu.com/en-us/inscribed-angle/inscribed-angle-06-02.png "Inscribed Angle: Example 1, Solution")
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.](https://ximpledu.com/en-us/inscribed-angle/inscribed-angle-07-02.png "Inscribed Angle: Example 2, Solution")
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.](https://ximpledu.com/en-us/inscribed-angle/inscribed-angle-08-02.png "Inscribed Angle: Example 3, Solution")
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.
