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Theorems of Finding Angle & Arc Measures

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson explains how to find angle and arc measures. The following theorems are discussed: tangent and intersected chord theorem, angles inside a circle theorem and angles outside a circle theorem.

Angle and Arc Measures in Circles

Imagine that you are trying to design a logo for a new company you are creating. You want to use circles and lines to create your logo. As you're a perfectionist, you want to make sure you are using just the right angles and arcs in your logo. Did you know that there is a specific relationship between the angles and arcs that are formed by lines and circles?

We can find the measure of angles that are formed inside, outside and on a circle if we know the arc measures. Conversely, we can also find the measure of arcs if we know certain angles that are formed inside, outside or on a circle.

Key Terms

Lines that are drawn in and through circles have specific names. You may recall that a radius is the length of a line drawn from the center of a circle to a point on the circle. You may also recall that a diameter is a line segment that is drawn from one point on a circle to another point but goes through the center. Other lesser-known lines include tangents, secants and chords.

Have you ever heard someone say that they went off on a tangent? Well that's because if a circle represents your train of thought, and you leave your train of thought and start talking about something else, you've gone off on a tangent. A tangent is a line that intersects a circle at exactly one point.


Tangent
Tangent


A line that is drawn through a circle is called a secant. A secant can have one end point on a circle, with the other end of the line continuing through the circle. A secant can also go all the way through a circle with no end points.


Secant
secant


A segment that has endpoints on a circle is called a chord. A chord can be drawn anywhere inside a circle. If the chord goes through the center of a circle, then it is called a diameter.


Chord
chord


Each of these lines can be used to create angles and arcs in a circle. There are specific rules for finding angle and arc measures, depending on where the angles are drawn and the lines used to draw them. The lines create intercepted arcs, which are the arcs formed by chords, tangents or secants. In this image, AB is the intercepted arc because it is intercepted by chords AC and CB.


Intercepted Arc
intercepted arc


Angles Formed Inside a Circle

Angles that are formed inside of a circle by two chords create four arcs on a circle. The measure of the angle is equal to half the sum of the intercepted arcs.


inside


The angle x is equal to half the sum of the intercepted arcs. The intercepted arc a is the arc from C to D. The intercepted arc b is the arc from A to B. To find the angle, we add the arcs and divide by 2.


inside formula


Example

Find the length of the angle x.


example 1


Since this angle is inside of a circle formed by two chords, we will add the arcs and divide by 2.


formula


answer


Angles Formed Outside a Circle

Angles that are formed outside of a circle can be formed in three ways:

  • Two tangents
  • Two secants
  • Tangent and a secant


Two Tangents
two tangents


Two Secants
two secants


Tangent and Secant
tangent & secant


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