Inscribed Angle

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.

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

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

Proof

Inscribed Angle: Proof - Case 1: The center of the circle is on the side of an inscribed angle.

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].

Inscribed Angle: Proof - Case 2: The center of the circle is in the interior of an inscribed angle.

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].

Inscribed Angle: Proof - Case 3: The center of the circle is in the exterior of an inscribed angle.

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

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

m[blue arc] = 80

x = (1/2)⋅80

Example 2

Find the measure of arc ABC. The measure of angle APC: 100.

m∠APC = 100

100 = (1/2)⋅m[arc ABC]

Example 3

Find the value of x. The measure of the inscribed angle: 7x + 1, 3x + 29.

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

So 7x + 1 = 3x + 29.