### Video Transcript

Given that the measure of angle
π΄π΅π· equals 44 degrees and the measure of angle πΆπΈπ΄ equals 72 degrees, find π₯,
π¦, and π§.

Letβs start by listing what we
know. Angle π΄π΅π· equals 44 degrees,
angle πΆπΈπ΄ measures 72 degrees. These two chords intersect at point
πΈ. And that means we can say that
angle π΅πΈπ· and angle πΆπΈπ΄ are vertical angles, which means their measure will be
equal to one another. They are congruent angles. And in this case, that means that
angle π΅πΈπ· is also equal to 72 degrees. The points πΈ, π΅, and π· form a
triangle, which means that their three angles must sum to 180 degrees. And we can substitute what we know
for these three angles into this equation. 72 plus 44 equals 116. 116 plus π§ equals 180. So, we subtract 116 from both
sides. And we find that π§ equals 64
degrees.

We wonβt be able to follow the same
procedure to find π₯ and π¦. So, weβll need to think about some
of the circle theorems. If we look at inscribed angle π΅,
we see that it has endpoints along the circle at π΄ and π· and that its intercepted
arc is arc π΄π·. We could write them as arc π΄π·
intercepts angle π΄π΅π·. But thereβs another angle in this
circle that also intercepts the same arc, and that would be angle π΄πΆπ·. Because both of these angles are
subtended by the same arc, we can say that the measure of angle π΄πΆπ· will be equal
to the measure of angle π΄π΅π·. And that means π₯ will be equal to
44 degrees.

And because all three angles need
to sum to 180 degrees, we can tell that angle π¦ is going to be equal to 64
degrees. If we wanted to confirm this, we
could see that angle πΆπ΄π΅ intercepts arc πΆπ΅ and angle πΆπ·π΅ intercepts arc
πΆπ΅. And so, we found that π₯ equals 44
degrees, and both π¦ and π§ equals 64 degrees.