Conjectures In Geometry Inscribed Angles In Circles ( Read )

You’re going to learn how to use the properties of these angles and intercepted arcs to find angle measurements.

Đang xem: Geometry inscribed angles

So let’s dive in!

Inscribed Angle Theorem

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of a circle.

This is different than the central angle, whose vertex is at the center of a circle. If you recall, the measure of the central angle is congruent to the measure of the minor arc.

However, when dealing with inscribed angles, the Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of the intercepted arc.

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Inscribed Angle Theorem

This means we can find the arc if we are given an inscribed angle, or we can find an inscribed angle if we know the measure of its intercepted arc.

Moreover, if two inscribed angles of a circle intercept the same arc, then the angles are congruent.

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Now a cool result of the theorem is that an angle inscribed in a semicircle is a right angle.

Why?

Because a semicircle (half a circle) creates an intercepted arc that measures 180°, therefore, any corresponding inscribed angle would measure half of it, as Varsity Tutors nicely states.

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Inscribed Polygon

Additionally, if all the vertices of a polygon lie on a circle, then the polygon is inscribed in the circle, and the circle is circumscribed about the polygon.

And if a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary.

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Inscribed Quadrilateral Theorem

How To Solve Inscribed Angles

In the diagram below, we are given a circle where angle ABC is an inscribed angle, and arc AC is the intercepted arc.

Using the theorem, we can quickly solve for either the inscribed angle or the arc. Notice how angle ABC is one-half the measure of the intercepted arc AC. Moreover, we can see that the intercepted arc AC is twice the measure of the inscribed angle.

Consequently, we can solve for both the inscribed angle or the intercepted arc using this one amazing theorem!

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Inscribed Angle Example

In the video below you’re going to learn how to find the measure of indicated angles and arcs as well as create systems of linear equations to solve for the angles of an inscribed quadrilateral.

Video – Lesson & Examples

38 min

Introduction00:00:22 – Overview of the theorems and intercepted arcs (Examples #1-3)00:10:30 – Semicircles and the inscribed quadrilateral theorem (Examples #4-5)00:20:15 – Find the indicated measure given an inscribed angle, quadrilateral or semicircle (Examples #6-8)Practice Problems with Step-by-Step Solutions Chapter Tests with Video Solutions

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