Inscribed Angles and Polygons

A Lesson on a Webpage Created by Mr. A

Today's lesson is sort of in a "blog" format.  This "blog" post is meant to provide you instruction on inscribed angles within circles.  Please read through each section of vocabulary and problems below and, concurrently, fill out your notes guide via Google Docs.

Part I:  What are inscribed angles?

Inscribed angles are angles whose vertex is on a circle and whose sides contain chords of the circle.  The rays of an inscribed angle intersect the circle in an intercepted arc.  In the diagram below, the angle ABC is the inscribed angle (its vertex is a point on the circle) and minor arc AC is the intercepted arc.

Part II:  The Measure of an Inscribed Angle

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.  Conversely, the intercepted arc's degree measure is twice that of the measure of an inscribed angle.  See the diagram below for the formula of the measure of an inscribed angle (include the diagram and formula in your HDN).

Examine the following examples of inscribed angle measures and their solutions.  Take a look at each problem individually, attempt the problem yourself, and then check the solutions.

Part III:  Congruent Inscribed Angles

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.  In this case, two inscribed angles have different vertices along the circle's locus, however the two inscribed angles share an arc.  In this case, the inscribed angles that share an arc must be congruent.  Examine this case in the examples below.

Part IV:  Inscribed Polygons

If all of the vertices of a polygon lie on a circle's locus, then the polygon is inscribed in the circle and the circle is circumscribed about the polygon.  This is displayed in the image below.

The polygons are inscribed in the circles. The circles are circumscribed about the polygons.

We have two specific theorems involving inscribed angles and quadrilaterals.  Both theorems and diagrams (click the "forward" arrow in the box below to see the other diagram) are to be placed in your HDN.  These are theorems are:

  • An inscribed angle inscribed within a semicircle is a right angle.  Additionally, a right triangle inscribed in a circle has the diameter as its hypotenuse.
  • A quadrilateral is inscribed in a circle if and only if its opposite angles are supplementary.

The examples posted below practice problems involving the above two theorems.  Attempt to solve each problem on your own first before scrolling on to the solutions.

Congratulations! You have reached the end of this lesson.  You may now complete the practice problems on your notes sheet in Google Docs and share your completed document with me using email address

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