Theorems of Finding Angle & Arc Measures

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  • 0:04 Angle & Arc Measures…
  • 0:37 Some Key Terms
  • 2:06 Angles Formed Inside a Circle
  • 2:55 Angles Formed Outside a Circle
  • 3:35 Tangent & Intercepted Arcs
  • 3:59 Lesson Summary
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Lesson Transcript
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 & Arc Measures in Circles

Let's start this lesson by trying to imagine that you're trying to design a logo for a new company you're creating. You want to use circles and lines to create your logo. As you're a perfectionist, you want to make sure you're 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.

Some 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's 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.


Another definition we have to look at is the line that's drawn through a circle, which 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.


Then there's a segment that has endpoints on a circle, which 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's called a diameter.


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's 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, which you can see in this diagram. The measure of the angle is equal to half the sum of the intercepted arcs.


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, like you can see in this formula.

inside formula

Here's an 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.



This, in turn, gives us our answer, which (as you can see here) is 145 degrees.

Angles Formed Outside a Circle

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

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