# Inscribed Angles and Arcs including Circumscribed and Inscribed Polygons.

### The focus of this lesson is in presenting to you how to give the right answers to these type of questions. We start by defining inscribed angles in terms of the angular measure of the arc they intersect, and we continue to present a multi step problem that involves inscribed polygons with inscribed angles, and central angles intercepting arcs in the circle. ¡You will find yourself increasingly able to work problems with circles!

Lesson's Content

 Lesson In PDF Format (no animations)

Lesson's Glossary

Arc
The curved segment that is between two points in the circumference of a circle.

Arc length
The distance between an arc's endpoints along the path of the circle.

Center of a circle
The point that all points in the circle are equidistant from.

Central angle of a circle
An angle whose vertex is the center of the circle.

Circumscribed circle or circumcircle of a polygon: A polygon is circumscribed about a circle if all the sides are tangent to the circle, and a circle is circumscribed about a polygon if all the vertices of this are on the circumference of the circle.

Chord of a circle
A segment whose endpoints are on a circle.

Circle
The set of points on a plane at a certain distance (radius) from a certain point (center); a polygon with infinite sides.

Inscribed angle
An inscribed angle is an angle whose vertex is on a circle and whose rays intersect the circle.

Inscribed planar shape or solid: A polygon is inscribed in a circle if the vertices of a polygon inside a circle are on the circumference of the circle; a circle is inscribed to a polygon if all the sides of the polygon are tangent to the circle in the interior of the polygon.

Minor arc
An arc whose endpoints form an angle less than 180 degrees with the center of the circle.

Major arc
An arc whose endpoints form an angle over 180 degrees with the center of the circle.

The segment whose endpoints are any point on a circle or sphere and its center; the length of that segment.

Sector
Part of a circle containing its center and an arc.

Semicircle
An arc whose central angle is a right angle.

Interactive Geometric Applets: Relevant Theorems.

This figure has an inscribed angle and a central angle intersecting the same arc in the

circle. You may drag any of the points "A" or "B" in the figure to verify the existing relationship:

The central angle is twice the measure of the inscribed angle intersecting the same arc.

Two inscribed angles intersecting the congruent arcs in the same circle or in congruent circles

are congruent. You may play with this interactive applet by dragging the vertices of the

inscribed angles, or the endpoints in the arc. Observe how the angles remain congruent.

Vocabulary Puzzle Interactive

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