### Video Transcript

Find the measure of ๐ฆ.

Letโs look at the diagram carefully. It consists of a circle with two chords, the lines ๐ด๐ต and ๐ถ๐ท. ๐ฆ is one of the angles formed at the point, where these two chords intersect.

Weโve also been given the measures of two of the arcs in the circles. The measure of the minor arc ๐ต๐ท is 77 degrees. And the measure of the minor arc ๐ด๐ถ 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 weโve been given are ๐ด๐ถ and ๐ต๐ท, which means the included angle here is not in fact the angle ๐ฆ, but itโs this angle that Iโve marked as angle ๐ฅ.

In order to calculate the measure of angle ๐ฆ directly using this result, we would need to know the measures of the arcs ๐ต๐ถ and ๐ด๐ท, which we donโt. However, angle ๐ฆ lies on a straight line with angle ๐ฅ. 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 ๐ฅ and then use the angle sum of angles ๐ฅ and ๐ฆ to find angle ๐ฆ. 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 ๐ฅ is half the sum of 77 degrees and 83 degrees. So the measure of angle ๐ฅ is 160 degrees divided by two, which is 80 degrees.

Now we know the measure of ๐ฅ, we can apply the fact that angles on a straight line sum to 180 degrees in order to calculate the value of ๐ฆ. The measure of angle ๐ฆ 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.