Video Transcript
In rectangle 𝐴𝐵𝐶𝐷, line segment 𝑀𝑁 is parallel to line segment 𝐴𝐷 and line segment 𝐵𝐶 and 𝑀𝐷 equals 𝑀𝐶. If the measure of angle 𝐷𝑀𝑁 equals 30 degrees, find the measure of angle 𝐶𝑀𝑁.
We can start by adding the extra information we are given to the diagram, with the three parallel line segments 𝑀𝑁, 𝐴𝐷, and 𝐵𝐶. We also have that the line segments 𝑀𝐷 and 𝑀𝐶 are congruent. So, given that the measure of angle 𝐷𝑀𝑁 is 30 degrees, we need to find the measure of angle 𝐶𝑀𝑁.
Let’s return to the information that 𝑀𝐷 equals 𝑀𝐶. These congruent line segments form two sides of triangle 𝑀𝐶𝐷. And by recalling that an isosceles triangle is a triangle that has two congruent sides, we can recognize that triangle 𝑀𝐶𝐷 is an isosceles triangle.
Now, there are quite a few properties of isosceles triangles that may be useful here. And this is particularly true with regards to the line segment 𝑀𝑁, which passes through the triangle. If we knew for sure that line segment 𝑀𝑁 was either a median or perpendicular to line segment 𝐷𝐶, that would give us some important information about the angle at the vertex of the isosceles triangle. So let’s see if we can prove either of these.
We can return to the fact that 𝐴𝐵𝐶𝐷 is a rectangle. And we know that rectangles have four congruent angles. All the angles at the vertices will have measures of 90 degrees. Therefore, because line segments 𝐴𝐷 and 𝐷𝐶 have a 90-degree angle measure at angle 𝐴𝐷𝐶, we can say that they are perpendicular. And therefore, we can note that line segments 𝐷𝐶 and 𝑀𝑁 must also be perpendicular, since we can use the property that if a line is perpendicular to any line 𝑋𝑌, then it is also perpendicular to any line parallel to line 𝑋𝑌.
Thus, we have demonstrated that the line segment 𝑀𝑁, which cuts through the isosceles triangle, is perpendicular to the base, the line segment 𝐷𝐶. We can then apply the following property. The straight line that passes through the vertex angle of an isosceles triangle and is perpendicular to the base bisects the vertex angle. The line segment 𝑀𝑁 does pass through the vertex angle, and it is perpendicular to the base. Therefore, the vertex angle is bisected.
The vertex angle in triangle 𝑀𝐶𝐷 is angle 𝐷𝑀𝐶. So the two angles within it, that’s angles 𝐶𝑀𝑁 and 𝐷𝑀𝑁, must have equal measures. We were given that the measure of angle 𝐷𝑀𝑁 is 30 degrees. So the measure of angle 𝐶𝑀𝑁 is the same. Therefore, we can give the answer that the measure of angle 𝐶𝑀𝑁 is 30 degrees.
