Question Video: Determining Whether a Given Triangle Is Isosceles Mathematics

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.