# Question Video: Finding the Measure of a Central Angle given Its Arc’s Measure Using Another Inscribed Angle’s Measure Mathematics

Find 𝑚∠𝐴𝐵𝐶.

02:55

### Video Transcript

Find the measure of angle 𝐴𝐵𝐶.

Let’s begin by identifying the angle whose measure we’re asked to find. It’s the angle formed when we move from 𝐴 to 𝐵 to 𝐶. So that’s this angle here on the figure. Now this is an inscribed angle on the circle’s circumference. So we know that its measure will be one-half of its intercepted arc. Its intercepted arc is the arc 𝐴𝐶. So we have the equation the measure of the angle 𝐴𝐵𝐶 is equal to one-half the measure of the arc 𝐴𝐶.

Let’s consider then how we might be able to calculate the measure of this arc. We can see that the other information given in the question is, firstly, the angle formed by the intersection of two chords inside a circle, the chords 𝐴𝐵 and 𝐶𝐷. We’re also told the measure of the arc intercepted by this angle, the measure of the arc 𝐵𝐷, which is 98 degrees. The angles of intersecting chords theorem tells us that the measure of the angle between two chords that intersect inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. The arc intercepted by the angle of 88 degrees is the arc 𝐵𝐷, and the arc intercepted by its vertical angle is the arc 𝐴𝐶. So we can form an equation 88 degrees is equal to one-half the measure of the arc 𝐵𝐷 plus the measure of the arc 𝐴𝐶.

Remember though that we know the measure of the arc 𝐵𝐷. It’s given to us in the figure as 98 degrees. So we can substitute this value into our equation, and we’ll then be able to solve to find the measure of the arc 𝐴𝐶. We have 88 degrees is equal to one-half of 98 degrees plus the measure of the arc 𝐴𝐶. Multiplying each side of the equation by two, we have 176 degrees equals 98 degrees plus the measure of the arc 𝐴𝐶. And finally, subtracting 98 degrees from each side, we find that the measure of the arc 𝐴𝐶 is 78 degrees.

The final step in this problem is to take this value for the measure of the arc 𝐴𝐶 and substitute it into our first equation. We have then that the measure of the angle 𝐴𝐵𝐶, which is half the measure of its intercepted arc, is one-half multiplied by 78 degrees, which is 39 degrees. So, by recalling the relationship between the measures of an inscribed angle and its intercepted arc and also the angles of intersecting chords theorem, we’ve found that the measure of the angle 𝐴𝐵𝐶 is 39 degrees.