### Video Transcript

Find ๐ฅ.

Letโs look at the diagram weโve been given. There is a circle and then two lines ๐ด๐ต and ๐ด๐ถ which are ach tangents to the circle because they each intercept the circle in only one place. These two tangents intersect at a point outside the circle, point ๐ด. And weโre told that the measure of the angle formed by their intersection is ๐ฅ. The other piece of information weโre given in the diagram is the measure of the minor arc ๐ต๐ถ. Thatโs the minor arc intercepted by these two tangents.

In order to calculate the value of ๐ฅ, we need to recall the angles of intersecting tangents theorem. This states that the measure of the angle formed by the intersection of two tangents outside a circle is half the positive difference of the measures of the intercepted arcs. We already said that the minor arc intercepted by these two tangents is the arc ๐ต๐ถ, whose measure weโve been given. The major arc intercepted by these two tangents is the major arc ๐ต๐ถ. And in order to distinguish between these two arcs, we can place a point ๐ท anywhere on the major arc. To obtain the positive difference, we need to subtract the measure of the minor arc from the measure of the major arc. So we have that ๐ฅ is equal to a half the measure of the arc ๐ต๐ท๐ถ minus the measure of the arc ๐ต๐ถ.

Weโre given the measure of the arc ๐ต๐ถ, but what about the arc ๐ต๐ท๐ถ? We can work this out if we recall that the measure of a full circle is 360 degrees and the measures of the major and minor arcs must therefore sum to this value. Subtracting 151 degrees from 360 degrees then, we find that the measure of the major arc ๐ต๐ท๐ถ is 209 degrees. We have then that ๐ฅ is equal to a half of 209 degrees minus 151 degrees. Thatโs a half multiplied by 58 degrees, which is 29 degrees. So by recalling the angles of intersecting tangents theorem, we found that the value of ๐ฅ, which is the measure of the angle formed by the intersection of two tangents outside a circle, is 29 degrees.