Central Angles and Intercepted Arcs - Problem 3

In this problem we’re being asked to find the measure of arc X. so X is the measure of the arc between those two points. What do we know? Well we know that this chord is congruent to that chord because they have the same number of congruence marks. Well, how does that help us?

Well, to do that we’re going to take a look at what we know about congruent chords. Well they do two things, they have congruent central angles and their intercepted arcs are also congruent. So we can use that to find the measure of this arc right here which has to be congruent to 120 degrees. As I remember from the beginning of geometry, a circle is 360 degrees. So if I add up 80 degrees plus 120 degrees plus X plus 120 they need to sum to 360 degrees because that’s a full circle.

So if I add this up, 80 and 120 is 200 plus 120 is 220, excuse me 200, 320. So we find that X when we subtract 320 from both sides is going to be 40 degrees. So the measure of arc X is 40 degrees, which we found by remembering that congruent chords have congruent arcs.