Central Angles and Intercepted Arcs - Problem 2

In this problem we’re being asked to find two different things; the measure of arc AB, so that’s going to be the measure of the arc between points A and B, and the measure of angle C.

Well, let’s start off by saying the measure of arc AB is going to be equal to the central angle that forms it. So AB is 140 degrees. We know that from our statement that the measure of a central angle is equal to its intercepted arc.

Now let’s try and figure out what angle C is. If I look closely, I see that I have two radii, and I also see that this is a linear pair with 140 degrees, which means this angle right here must be 40 degrees. Since I have an isosceles triangle, I could call both of these angles X.

So I can write the statement that 40 plus 2X must be 180 degrees. So I would solve for X by subtracting 40, so 2X equals 140 degrees, and I’m going to divide by 2. So X must be 70 degrees, and I labeled X as my angle C. So C must be 70 degrees. The key thing there was realizing that we have two radii which are always congruent within the same circle.