Video Transcript
In the figure below, the
intersection of line segment 𝐴𝐶 and line segment 𝐵𝐷 equals 𝑀. 𝑀𝐷 equals 𝑀𝐶. Line segment 𝐴𝐵 and line segment
𝐶𝐷 are parallel. And the measure of angle 𝑀𝐶𝐷
equals 40 degrees. Draw line segment 𝑀𝐸
perpendicular to line segment 𝐶𝐷, and cut line segment 𝐶𝐷 at 𝐸. Then, draw line segment 𝑀𝐹
perpendicular to line segment 𝐴𝐵, and cut line segment 𝐴𝐵 at 𝐹. Find the measure of angle 𝐶𝑀𝐸
and the measure of angle 𝐵𝑀𝐹.
Let’s begin by considering the
information that we are given, with the first piece of information being that the
two line segments 𝐴𝐶 and 𝐵𝐷 intersect at the point 𝑀. Next, we have two congruent line
segments, since 𝑀𝐷 is equal to 𝑀𝐶. Then, we are given that the line
segments 𝐴𝐵 and 𝐶𝐷 are parallel. And the measure of angle 𝑀𝐶𝐷 is
40 degrees.
Now we need to draw two new line
segments, with the first being the line segment 𝑀𝐸, which must be perpendicular to
line segment 𝐶𝐷 and with line segment 𝑀𝐸 cutting line segment 𝐶𝐷 at point 𝐸,
which would look like this on the figure. Similarly, we can draw the second
new line segment of 𝑀𝐹 perpendicular to line segment 𝐴𝐵, which cuts line segment
𝐴𝐵 at the point 𝐹. We will need to determine the
measures of angles 𝐶𝑀𝐸 and 𝐵𝑀𝐹.
One way that we can find the
measure of the first angle 𝐶𝑀𝐸 is by noticing that we have triangle 𝐶𝑀𝐸 and we
know two of the angles within it. By recalling that the interior
angle measures in a triangle sum to 180 degrees, we can write that 40 degrees plus
the right angle of 90 degrees plus the measure of angle 𝐶𝑀𝐸 equals 180
degrees. And simplifying the left-hand side
and then subtracting 130 degrees from both sides, we have that the measure of angle
𝐶𝑀𝐸 is 50 degrees. So, that’s the first angle measure
calculated.
Next, let’s consider how we can
calculate the measure of angle 𝐵𝑀𝐹. To do this, we can take a closer
look at triangle 𝐶𝑀𝐷. Given that this triangle has two
congruent line segments, that means that by definition, triangle 𝐶𝑀𝐷 is an
isosceles triangle. We also drew a perpendicular line
from the vertex angle of this isosceles triangle to the base.
We can recall that one of the
corollaries of the isosceles triangle theorems states that the straight line that
passes through the vertex angle of an isosceles triangle and is perpendicular to the
base bisects the base and the vertex angle. The important part for this
question is that the line segment 𝑀𝐸 that we drew bisects the vertex angle. The vertex angle would be the angle
𝐶𝑀𝐷. So both angles 𝐶𝑀𝐸 and 𝐷𝑀𝐸
would have the same measure. They are both 50 degrees.
But we still need to find the
measure of angle 𝐵𝑀𝐹, which is in a different triangle. However, we can observe that the
angles 𝐷𝑀𝐸 and 𝐵𝑀𝐹 are vertically opposite angles. Therefore, they are both equal in
measure. They are both 50 degrees. We can then give the answers for
both the required angle measures. The measure of angle 𝐶𝑀𝐸 is 50
degrees, and the measure of angle 𝐵𝑀𝐹 is 50 degrees.
