### Video Transcript

Given that the measure of angle
π΄π΅πΆ equals 27 degrees and the measure of angle π΅π·π΄ equals 62 degrees, find the
measure of angle π΅π·πΆ and the measure of angle π΅πΆπ·.

The first angle measure that weβre
given is the measure of angle π΄π΅πΆ, which is 27 degrees. The second angle measure is that of
angle π΅π·π΄, which weβre told is 62 degrees. The first angle weβre asked to
calculate the measure of is that of angle π΅π·πΆ, which is the whole angle at the
top of this figure. The second angle we need to
calculate is that of angle π΅πΆπ·, which is on the right of the figure.

The first thing we might observe is
that we have this bow tie shape within the circle, which may indicate that weβll be
using inscribed angles theorems. One of these theorems tells us that
inscribed angles subtended by the same arc are equal. This angle of πΆπ΅π΄ is inscribed
by the arc πΆπ΄. And so it will be equal in measure
to any other angle inscribed by the same arc. And so the measure of angle πΆπ·A
will be the same size. It will be 27 degrees. The first angle we needed to
calculate was the measure of angle π΅π·πΆ. And so it will be the sum of these
two angles, 62 degrees plus 27 degrees, which gives us an answer of 89 degrees. And so thatβs the first angle
measure found.

Now, letβs see how we work towards
finding the other angle measure. Because we know that inscribed
angles subtended by the same arc are equal, then the measure of angle π·π΄π΅ will be
equal to the measure of angle π·πΆπ΅. However, we donβt know the measure
of this angle either. Letβs see if we can calculate
either of these angles. To do this, letβs observe that we
have a pair of congruent line segments. The line segment π΄π· is equal to
the line segment π΄π΅. And so when we consider these line
segments as part of the triangle π΄π΅π·, then we know that π΄π΅π· will be an
isosceles triangle.

Isosceles triangles have two equal
sides and two equal angle measures. And so the two angle measures that
are equal will be the measure of angle π΄π·π΅ and the measure of angle π΄π΅π·. These will both be 62 degrees. We can now use the triangle π΄π΅π·
and the fact that the interior angles in a triangle sum to 180 degrees to find the
measure of angle π·π΄π΅. We can write the equation that the
three angles in the triangle, thatβs 62 degrees, 62 degrees, and the measure of
angle π·π΄π΅, must sum to 180 degrees. 62 degrees plus 62 degrees
simplifies to 124 degrees. Subtracting 124 degrees from both
sides gives us that the measure of angle π·π΄π΅ is 56 degrees.

Now, we know that this angle
subtended by the arc π΅π· will be equal to the measure of angle π΅πΆπ·, which is
subtended by the same arc. And so we can give the answer for
both angle measures. The measure of angle π΅π·πΆ is 89
degrees, and the measure of angle π΅πΆπ· is 56 degrees.