Video Transcript
Is 𝑀𝐵𝐶 an isosceles
triangle?
Let’s begin by recalling that an
isosceles triangle is a triangle that has two congruent sides. Now we might notice in this figure
that there are two congruent sides marked. Line segments 𝐴𝐶 and 𝐴𝐵 are
congruent. But be careful, because this tells
us that triangle 𝐴𝐵𝐶 is an isosceles triangle. And we need to determine if
triangle 𝑀𝐵𝐶 is isosceles. So let’s see if we can work out
either some side lengths or some angles in this smaller triangle. And to help us, we can use the fact
that triangle 𝐴𝐵𝐶 is isosceles.
We know that in an isosceles
triangle, the angles opposite the congruent sides are congruent. These are the base angles in the
isosceles triangle. In triangle 𝐴𝐵𝐶, these base
angles will be the angles 𝐴𝐶𝐵 and 𝐴𝐵𝐶. Let’s see if we can find the
measures of these angles. In any triangle, the angle measures
sum to 180 degrees, so we know that the three angle measures in triangle 𝐴𝐵𝐶 —
that’s the measure of angle 𝐴𝐶𝐵, the measure of angle 𝐴𝐵𝐶, and the measure of
angle 𝐶𝐴𝐵 — will all sum to 180 degrees. And we were given that the measure
of angle 𝐶𝐴𝐵 is 44 degrees. So, by subtracting 44 degrees from
both sides, we have that the measure of our two unknown angles is equal to 136
degrees.
And we’ve already noted that these
two angles are congruent, that is, that the measure of angle 𝐴𝐵𝐶 is equal to the
measure of angle 𝐴𝐶𝐵. So we can write that two times the
measure of angle 𝐴𝐶𝐵 is 136 degrees. By dividing through by two, we have
that the measure of angle 𝐴𝐶𝐵 is 68 degrees. That means that both angles 𝐴𝐶𝐵
and 𝐴𝐵𝐶 have a measure of 68 degrees.
Now let’s consider the triangle
𝑀𝐵𝐶 and, in particular, the angle 𝑀𝐶𝐵. Its measure will be equal to 68
degrees minus 37 degrees, which is 31 degrees. Notice that we were given that the
measure of angle 𝑀𝐵𝐶 is 31 degrees, so triangle 𝑀𝐵𝐶 has two congruent angles
of 31 degrees each. The converse of the isosceles
triangle theorem states that if two angles in a triangle are congruent, then the
sides opposite those angles are congruent. And by definition, we would have an
isosceles triangle.
Therefore, we can give the answer
to the question as yes, since we have proved that triangle 𝑀𝐵𝐶 is an isosceles
triangle.
