### Video Transcript

Given that the line segment π΄π΅ is
a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅,
find the measure of angle π΄πΈπ·.

Weβre interested in the measure of
angle π΄πΈπ·; thatβs this measure. And weβve been given a few other
pieces of information. We know line segment π·πΆ is
parallel to line segment π΄π΅. We know line segment π΄π΅ is the
diameter. And on the figure, angle πΆπ΅π΄ has
been labeled as 68.5 degrees.

At first, it might not seem like
thereβs a very clear direction for where to go here. But if we start with the measure of
angle πΆπ΄π΅, using that information, we could find the measure of arc πΆπ΄. Since angle πΆπ΄π΅ is an inscribed
angle, its arc will be two times the measure of that inscribed angle. Arc π΄πΆ will then be equal to two
times 68.5, which is 137 degrees. And because we know that line
segment π΄π΅ is a diameter, arc π΄π΅ must be equal to 180 degrees. We can also say that arc π΄π΅ will
be equal to arc π΅πΆ plus arc πΆπ΄.

We know π΄π΅ needs to be 180
degrees and arc πΆπ΄ is 137 degrees. To solve for the measure of arc
π΅πΆ, we can subtract 137 from both sides of the equation. And we get the measure of arc π΅πΆ
is 43 degrees. And hereβs where our parallel
chords come into play. When you have parallel chords,
their intercepted arcs are going to be congruent. And that means because arc πΆπ΅
equals 43 degrees, arc π·π΄ also equals 43 degrees.

And at this point, we began to see
that arc π·π΄ is subtended by the angle π΄πΈπ·. Since angle π΄πΈπ· is an inscribed
angle, its angle measure, the measure of angle π΄πΈπ·, is going to be equal to
one-half the measure of arc π΄π·. We know that the measure of arc
π΄π· is 43 degrees, and one-half of 43 is 21.5. And so, we can say that the measure
of angle π΄πΈπ· is 21.5 degrees.