Question Video: Finding the Angle of a Point outside a Circle given the Angle of the opposite Arc Mathematics

Video Transcript

Find 𝑥.

In this question, we’re asked to find the value of 𝑥. And we can see in our diagram that 𝑥 is the angle between two secant lines which intersect outside of our circle. And we can find the measure of 𝑥 by recalling the following fact. The angle between two secant lines in a circle which intersect outside of a circle is one-half the positive difference of the measures of the arcs intercepted by the sides of the angle.

To apply this property, let’s do this step by step. First, let’s mark the sides of the angle of 𝑥. We can see that 𝑥 is the angle between the lines 𝐴𝐶 and 𝐴𝐸. So the two sides of our angle are the line segment 𝐴𝐶 and the line segment 𝐴𝐸. Next, we need to find the measures of the arcs intercepted by the two sides of our angle. The first side of our angle intersects the circle at the point 𝐵, and the second side of our angle intersects the circle at the point 𝐷. So one of the arcs we’re going to use is the arc from 𝐵 to 𝐷. Similarly, the first side of our angle intercepts the circle at the point 𝐶, and the second side of our angle intercepts the circle at the point 𝐸. So the other arc we’re interested in is the arc from 𝐶 to 𝐸.

Finally, the measure of our angle will be one-half the positive difference between the measures of these two arcs. And since the arc from 𝐶 to 𝐸 is bigger than the arc from 𝐵 to 𝐷, this gives us the following result. 𝑥 will be equal to one-half multiplied by the measure of arc 𝐶𝐸 minus the measure of arc 𝐵𝐷. And we’re given both of these values in the diagram. The measure of arc 𝐶𝐸 is 132 degrees, and the measure of arc 𝐵𝐷 is 36 degrees. So we substitute these values into our formula. We get that 𝑥 is equal to one-half multiplied by 132 degrees minus 36 degrees. And we can then evaluate this expression. 132 minus 36 is equal to 96. And if we multiply this by one-half, we get 48. Therefore, 𝑥 is equal to 48 degrees.