### Video Transcript

In the following figure, if πΏπ
equals ππ, πΏπ equals ππ, the measure of angle πΏππ equals 30 degrees, the
measure of angle ππΏπ equals 55 degrees, and π is the midpoint of line segment
πΏπ, find the measure of angle πππΏ and the measure of angle πΏππ.

Letβs begin this question by noting
that we have three pairs of congruent line segments. Firstly, we have that line segments
πΏπ and ππ are congruent. Then, πΏπ and ππ are
congruent. And finally we can see on the
diagram that line segments πΏπ and ππ are congruent. Thatβs because the point π is the
midpoint of the line segment πΏπ.

Now, the first two pairs of
congruent sides will give us some information about these triangles. We can say that triangle πΏππ is
isosceles. And so is triangle πΏππ as both
these triangles have two sides congruent. Now, letβs consider the angle
measures that we need to calculate. The first is the measure of angle
πππΏ, which means that we need to add in a line segment. Here is the line segment ππ. So angle πππΏ will be here on the
left side of the diagram.

Notice that we could describe this
interior angle at π, that is, the angle πΏππ, to be the vertex angle of triangle
πΏππ, because this is an isosceles triangle. And in an isosceles triangle, the
angle which is not congruent to any other is called the vertex angle. And this information might help us
recall one of the isosceles triangle theorems. This theorem states that the median
of an isosceles triangle from the vertex angle is a perpendicular bisector of the
base and bisects the vertex angle.

Letβs remind ourselves of what a
median is. It is a line segment from any
vertex of a triangle to the midpoint of the opposite side. And thatβs what we have here. Line segment ππ is a line from
the vertex angle of triangle πΏππ to the midpoint of the opposite side. The theorem then allows us to note
that this median is a perpendicular bisector of the base πΏπ. Notice that we already knew that
the point π is the midpoint of line segment πΏπ. But now we know that the line
segment ππ bisects it at right angles.

By noticing that this angle is part
of the smaller triangle πΏππ, then we can find its measure by recalling that the
interior angle measures in a triangle sum to 180 degrees. So we have the measure of angle
πππΏ plus 55 degrees plus 90 degrees equals 180 degrees. Then, simplifying and subtracting
145 degrees from both sides, we have that the measure of angle πππΏ is 35
degrees. And thatβs the first part of the
question answered.

Next, we need to find the measure
of angle πΏππ. And so we need to draw in another
line segment. Here is the line segment ππ. And we can add in the angle πΏππ
whose measure we need to calculate. At this point, we canβt simply
assume that the larger line segment ππ is a straight line which passes through
π. So letβs see instead what we can
determine.

We can note that line segment ππ
is a median of triangle πΏππ. And as we previously noted,
triangle πΏππ is an isosceles triangle. Applying the same theorem as
before, then this median from the vertex angle of the isosceles triangle must be a
perpendicular bisector of the base. Therefore, the measure of angle
πΏππ is 90 degrees.

We can give the answers for both
the required angle measures. The measure of angle πππΏ is 35
degrees, and the measure of angle πΏππ is 90 degrees.