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

In the figure below, line segment 𝐴𝐢 ∩ line segment 𝐡𝐷 = 𝑀, 𝑀𝐷 = 𝑀𝐢, line segment 𝐴𝐡 βˆ₯ line segment 𝐢𝐷, and π‘šβˆ π‘€πΆπ· = 40Β°. Draw line segment 𝑀𝐸 βŸ‚ line segment 𝐢𝐷 and cut line segment 𝐢𝐷 at 𝐸, then draw line segment 𝑀𝐹 βŸ‚ line segment 𝐴𝐡 and cut line segment 𝐴𝐡 at 𝐹. Find π‘šβˆ πΆπ‘€πΈ and π‘šβˆ π΅π‘€πΉ.

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Video Transcript

In the figure below, the intersection of line segment 𝐴𝐢 and line segment 𝐡𝐷 equals 𝑀. 𝑀𝐷 equals 𝑀𝐢. Line segment 𝐴𝐡 and line segment 𝐢𝐷 are parallel. And the measure of angle 𝑀𝐢𝐷 equals 40 degrees. Draw line segment 𝑀𝐸 perpendicular to line segment 𝐢𝐷, and cut line segment 𝐢𝐷 at 𝐸. Then, draw line segment 𝑀𝐹 perpendicular to line segment 𝐴𝐡, and cut line segment 𝐴𝐡 at 𝐹. Find the measure of angle 𝐢𝑀𝐸 and the measure of angle 𝐡𝑀𝐹.

Let’s begin by considering the information that we are given, with the first piece of information being that the two line segments 𝐴𝐢 and 𝐡𝐷 intersect at the point 𝑀. Next, we have two congruent line segments, since 𝑀𝐷 is equal to 𝑀𝐢. Then, we are given that the line segments 𝐴𝐡 and 𝐢𝐷 are parallel. And the measure of angle 𝑀𝐢𝐷 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 𝐢𝐷 and with line segment 𝑀𝐸 cutting line segment 𝐢𝐷 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 𝐴𝐡, which cuts line segment 𝐴𝐡 at the point 𝐹. We will need to determine the measures of angles 𝐢𝑀𝐸 and 𝐡𝑀𝐹.

One way that we can find the measure of the first angle 𝐢𝑀𝐸 is by noticing that we have triangle 𝐢𝑀𝐸 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 𝐢𝑀𝐸 equals 180 degrees. And simplifying the left-hand side and then subtracting 130 degrees from both sides, we have that the measure of angle 𝐢𝑀𝐸 is 50 degrees. So, that’s the first angle measure calculated.

Next, let’s consider how we can calculate the measure of angle 𝐡𝑀𝐹. To do this, we can take a closer look at triangle 𝐢𝑀𝐷. Given that this triangle has two congruent line segments, that means that by definition, triangle 𝐢𝑀𝐷 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 𝐢𝑀𝐷. So both angles 𝐢𝑀𝐸 and 𝐷𝑀𝐸 would have the same measure. They are both 50 degrees.

But we still need to find the measure of angle 𝐡𝑀𝐹, which is in a different triangle. However, we can observe that the angles 𝐷𝑀𝐸 and 𝐡𝑀𝐹 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 𝐢𝑀𝐸 is 50 degrees, and the measure of angle 𝐡𝑀𝐹 is 50 degrees.

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