Let’s look carefully at the diagram we’ve been given. We have a circle and two chords, 𝐴𝐵 and 𝐶𝐷, which intersect at a point inside the circle. We’re asked to find the value of 𝑥, which is the measure of one of the angles formed by the intersection of these two chords.
We can therefore recall the angles of intersecting chords theorem. This tells us that the measure of the angle formed by the intersection of two chords inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. We need to identify these two arcs then. The arc intercepted by angle 𝑥 is the arc 𝐴𝐶 because the two lines which form the sides of angle 𝑥 intercept the circle at the points 𝐴 and 𝐶. The vertical angle is this angle here, and the arc intercepted by this angle is the arc 𝐵𝐷.
So we can state that 𝑥 is equal to one-half the measure of the arc 𝐴𝐶 plus the measure of the arc 𝐵𝐷. Looking at the diagram, we can see that we have been given both of these values. The measure of the arc 𝐴𝐶 is 73 degrees, and the measure of the arc 𝐵𝐷 is 133 degrees. So we have that 𝑥 is equal to a half of 73 degrees plus 133 degrees. That’s a half of 206 degrees, which is 103 degrees. So by recalling the angles of intersecting chords theorem, we found that the value of 𝑥 is 103 degrees.