### Video Transcript

Given that π is the center of the
circle, find the measure of angle πππ.

We will begin by finding angle
πππ on the diagram. We recall that when an angle is
named with three letters, the middle letter is the vertex of the angle. So in this case, our angle has
vertex π. In the given diagram of the circle
with center π, we have highlighted angle πππ in orange. Letβs now collect any pertinent
information that is given in the diagram of the circle. We see that angle πππ is given a
measure of 102 degrees. That is the only angle measure that
is given on this diagram. We also notice that chord π΄π΅ has
an equal length to chord π΄πΆ. Since π΄π equals ππ΅, we identify
π as the midpoint of chord π΄π΅. And given that πΆπ equals ππ΄, we
also conclude that π is the midpoint of chord π΄πΆ.

When we consider that we have two
chords of equal length in the given diagram, this reminds us of a theorem that says
that two chords of equal lengths in the same circle are equidistant from the center
of the circle. We also know that the distance of a
chord from the center of the circle is measured by the length of the segment from
the center that perpendicularly bisects the chord.

In this example, we have two chords
π΄π΅ and π΄πΆ that have equal lengths. And we recall that the
perpendicular bisector of a chord will always go through the center of the
circle. Since π and π are the midpoints
of the two chords and π is the center of the circle, the line segments ππ and
ππ must be the perpendicular bisectors of the two chords. This tells us that ππ is actually
the distance from chord π΄π΅ to the center of the circle and that ππ is the
distance from chord π΄πΆ to the center.

Since the two chords π΄π΅ and π΄πΆ
have equal lengths, they must also be equidistant from the center. And this tells us that ππ equals
ππ. Having marked ππ equal to ππ in
pink on the diagram, we will now clear some space.

Now, we turn our attention to
triangle πππ. Since two sides of this triangle
have equal lengths, triangle πππ is considered an isosceles triangle. We recall the isosceles triangle
theorem, which says if two sides of a triangle are congruent, then the angles
opposite those sides are congruent. Knowing that congruent means equal
in measure and having ππ equal to ππ means that the measure of angle πππ
equals the measure of angle πππ. This fact must be important given
that we are looking for the measure of angle πππ. So we mark these two angles as
congruent on the diagram.

Now we know quite a lot about the
three interior angles of triangle πππ. We know that the measure of angle
πππ is 102 degrees and that the measure of the other two angles are equal. We also recall that the interior
angles in a triangle sum to 180 degrees. In the case of triangle πππ,
this means that the measure of angle πππ plus the measure of angle πππ plus
the measure of angle πππ equals 180 degrees. We know that the measure of angle
πππ equals 102 degrees and also that the measure of angle πππ equals the
measure of angle πππ. So we will substitute these
expressions into the equation. This gives us 102 degrees plus the
measure of angle πππ plus the measure of angle πππ equals 180 degrees.

Since the measure of angle πππ
is added to itself, we can simplify this as two times the measure of angle
πππ. Then subtracting 102 degrees from
each side of the equation leads us to two times the measure of angle πππ equals
78 degrees. Finally, after dividing both sides
of the equation by two, we have our answer. The measure of angle πππ equals
39 degrees.

Itβs a good idea at the end to do a
quick computational check. We want to see if the three
interior angles 102 degrees, 39 degrees, and 39 degrees add up to the sum that we
expected. Having shown that triangle πππ
is isosceles and verifying the sum of the interior angles is 180 degrees, we are
confident in our final answer.