An inscribed angle in a circle is developed by two chords that have actually a common end point on the circle. This typical end point is the peak of the angle.

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Here, the circle with facility O has the inscribed angle ∠ABC. The other finish points than the vertex, A and also C define the intercepted arc AC⌢ the the circle. The measure of AC⌢is the measure up of its central angle. That is, the measure up of ∠AOC.

Inscribed edge Theorem:

The measure of an inscribed edge is half the measure up of the intercepted arc.

You are watching: How to find the measure of an inscribed angle

the is, m∠ABC=12m∠AOC.

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This leads to the corollary that in a circle any kind of two inscribed angles v the very same intercepted arcs room congruent.

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Here, ∠ADC≅∠ABC≅∠AFC.


Example 1:

Find the measure up of the inscribed angle ∠PQR.

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through the inscribed edge theorem, the measure of one inscribed edge is fifty percent the measure up of the intercepted arc.

The measure of the main angle ∠PORof the intercepted arc PR⌢ is 90°.

Therefore,

m∠PQR=12m∠POR                   =12(90°)                    =45°.


Example 2:

Find m∠LPN.

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In a circle, any kind of two enrolled angles v the exact same intercepted arcs room congruent.

Here, the inscribed angle ∠LMNand ∠LPNhave the same intercepted arc LN⌢.

So, ∠LMN≅∠LPN.

Therefore, m∠LMN=m∠LPN=55°.


An particularly interesting an outcome of the Inscribed angle Theorem is that an edge inscribed in a semi-circle is a ideal angle.

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In a semi-circle, the intercepted arc measures 180°and therefore any type of corresponding enrolled angle would measure fifty percent of it.


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