# How to Determine the Measure of an Angle whose Vertex Is Inside a Circle

An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article covers angles that have their vertex inside a circle—so-called *chord-chord **angle**s**. *The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

For example, check out the above figure, which shows you chord-chord angle *SVT*. You find the measure of the angle like this:

Look at the following figure:

Here’s a problem to show how the formula plays out:

To use the formula to find angle 1, you need the measures of arcs *MJ* and *KL*. You know the ratio of all four arcs is 1 : 3 : 4 : 2, so you can set their measures equal to 1*x*, 3*x*, 4*x*, and 2*x*. The four arcs make up an entire circle, so they must add up to 360°. Thus,

1*x* + 3*x* + 4*x* + 2*x* = 360

10*x* = 360

* ** **x* = 36

Now use the formula:

That's a wrap.