### Video Transcript

Given that line segment π΄π΅ is a
diameter in circle π and the measure of angle π΅ππ· equals 59 degrees, find the
measure of angle π΄πΆπ· in degrees.

Letβs put what we know into the
diagram. Angle π΅ππ· measures 59 degrees,
and weβre trying to find the measure of angle π΄πΆπ·. If we start with what we know about
angle π΅ππ·, since π΅ππ· has a vertex at the center of the circle, π΅ππ· is a
central angle. And because angle π΅ππ· is a
central angle, its subtended arc, arc π΅π·, also measures 59 degrees. Weβre also interested in angle
π΄πΆπ·. But angle π΄πΆπ· is not a central
angle. Itβs an inscribed angle because its
vertex is on the circumference of the circle, as are both endpoints.

The arc associated with angle
π΄πΆπ· would be arc π΄π·. We have a partial measurement for
this arc, but weβre missing the distance from π΄ to π΅. But because we know that π΄π΅ is a
diameter, it cuts the circle in half. And that means the measure of arc
π΄π΅ is 180 degrees. If arc π΄π΅ equals 180 and arc π΅π·
equals 59 degrees, we can say the measure of arc π΄π· is equal to the measure of arc
π΄π΅ plus the measure of arc π΅π·.

If we plug in what we know, the
measure of arc π΄π· is 239 degrees. Because angle π΄πΆπ· is an
inscribed angle and it has a subtended arc measure of 239 degrees, we can find out
the exact measure of angle π΄πΆπ·. The measure of the inscribed angle
π΄πΆπ· will be one-half its subtended arc, arc π΄π·. Since that arc is 239 degrees, we
take half of that and we get 119.5 degrees for the measure of angle π΄πΆπ·.