Question Video: Proving a Geometric Statement Using the Corollaries of the Isosceles Triangle Theorems Mathematics

In the figure below, line segment 𝐴𝐶 ∩ line segment 𝐵𝐷 = 𝑀, 𝑀𝐷 = 𝑀𝐶, line segment 𝐴𝐵 ∥ line segment 𝐶𝐷, and 𝑚∠𝑀𝐶𝐷 = 40°. Draw line segment 𝑀𝐸 ⟂ line segment 𝐶𝐷 and cut line segment 𝐶𝐷 at 𝐸, then draw line segment 𝑀𝐹 ⟂ line segment 𝐴𝐵 and cut line segment 𝐴𝐵 at 𝐹. Find 𝑚∠𝐶𝑀𝐸 and 𝑚∠𝐵𝑀𝐹.

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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.