### Video Transcript

Find ๐ฅ.

Letโs have a look at the diagram
weโve been given more closely. It consists of a circle and then
two lines ๐ด๐ถ and ๐ด๐ธ. Each of these lines intersect with
the circle in two places, which means the lines ๐ด๐ถ and ๐ด๐ธ are called
secants. The value of ๐ฅ, which weโve been
asked to find, is the measure of the angle formed where these two secants meet.

Weโve also been given the measures
of the two intercepted arcs formed by these two secants. The measure of the minor arc ๐ต๐ท
is 36 degrees. And the measure of the major arc
๐ถ๐ธ is 132 degrees. We need to recall the relationship
that exists between the angle formed by two secants and the measures of the two
intercepted arcs. We recall then that โif two secants
intersect outside a circle, the measure of the angle formed is half the difference
of the measures of the intercepted arcs.โ

Our two secants do intersect at a
point outside the circle. They intersect at the point ๐ด. So we could apply this result. The measure of the angle formed is
the value ๐ฅ that weโre looking to find. Weโre told that this is equal to
half the difference of the measures of the intercepted arcs. So this is a half of the measure of
the arc ๐ถ๐ธ minus the measure of the arc ๐ต๐ท, both of which weโve been given.

๐ฅ is therefore equal to a half of
132 degrees minus 36 degrees. 132 minus 36 is 96. So ๐ฅ is equal to a half of 96
degrees or a half times 96 degrees. A half of 96, which we can find by
dividing 96 by two, is 48.

So we found that the value of ๐ฅ,
which is the measure of the angle formed between the two secants ๐ด๐ถ and ๐ด๐ธ, is
48 degrees.