Circles: Angle Measures and Segment Lengths

Standard

G.C.2 Identify and describe relationships among inscribed angles, radii, and chords.

Goals for this section:

To be able to find the angle measure of two intersecting lines that touch/go through a circle.

How to measure an arc

If an angle is formed with its vertex at the center of a circle, then the angle measurement and the arc measurement are the same.

Theorem: Angles whose vertex is on the circle

If the vertex of an angle is on the circle (or the intersection of two lines is on the circle), then the angle is half of the intercepted arc.

Theorem: Intersecting Lines inside a circle

The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs.

Theorem: Intersecting lines outside a circle

The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.

Below are three cases in which we use this theorem. The lines must be touching the circle in some way:

  • Both lines touch two points on the circle (goes through).
  • One line touches one point on the circle (touches the edge of a circle but does not go through) and the other line touches two points on the circle (goes through the circle).
  • Both lines touch only one point on the circle (touches the edge of a circle but does not go through.


Keywords: circle, intersecting lines, angle, arc