Find the measure of 𝑦.
Let’s look at the diagram carefully. It consists of a circle with two chords, the lines 𝐴𝐵 and 𝐶𝐷. 𝑦 is one of the angles formed at the point, where these two chords intersect.
We’ve also been given the measures of two of the arcs in the circles. The measure of the minor arc 𝐵𝐷 is 77 degrees. And the measure of the minor arc 𝐴𝐶 is 83 degrees. We need to record the relationship that exists between the angles formed by intersecting chords and the measures of their opposite arcs.
The relationship is this: If two chords intersect at a point inside a circle, then the measure of the included angle equals half the sum of the measures of the two opposite arcs. Now the measures of the two arcs we’ve been given are 𝐴𝐶 and 𝐵𝐷, which means the included angle here is not in fact the angle 𝑦, but it’s this angle that I’ve marked as angle 𝑥.
In order to calculate the measure of angle 𝑦 directly using this result, we would need to know the measures of the arcs 𝐵𝐶 and 𝐴𝐷, which we don’t. However, angle 𝑦 lies on a straight line with angle 𝑥. And therefore the measures of these two angles sums to 180 degrees.
Therefore, a strategy for this question is going to be to apply the result to calculate angle 𝑥 and then use the angle sum of angles 𝑥 and 𝑦 to find angle 𝑦. So the result tells us that the measure of the included angle is half the sum of the measures of the two opposite arcs. Therefore, the measure of angle 𝑥 is half the sum of 77 degrees and 83 degrees. So the measure of angle 𝑥 is 160 degrees divided by two, which is 80 degrees.
Now we know the measure of 𝑥, we can apply the fact that angles on a straight line sum to 180 degrees in order to calculate the value of 𝑦. The measure of angle 𝑦 is equal to 180 degrees minus 80 degrees, which is 100 degrees.
So we have our answer to the problem, 100 degrees. Remember the key fact that we used in this question: If two cords intersect at a point inside a circle, then the measure of the included angle equals half the sum of the measures of the two opposite arcs.