Question Video: Determining Whether a Given Triangle Is Isosceles Mathematics • 11th Grade

Is 𝑀𝐡𝐢 an isosceles triangle?


Video Transcript

Is 𝑀𝐡𝐢 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 𝐴𝐢 and 𝐴𝐡 are congruent. But be careful, because this tells us that triangle 𝐴𝐡𝐢 is an isosceles triangle. And we need to determine if triangle 𝑀𝐡𝐢 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 𝐴𝐡𝐢 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 𝐴𝐡𝐢, these base angles will be the angles 𝐴𝐢𝐡 and 𝐴𝐡𝐢. 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 𝐴𝐡𝐢 β€” that’s the measure of angle 𝐴𝐢𝐡, the measure of angle 𝐴𝐡𝐢, and the measure of angle 𝐢𝐴𝐡 β€” will all sum to 180 degrees. And we were given that the measure of angle 𝐢𝐴𝐡 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 𝐴𝐡𝐢 is equal to the measure of angle 𝐴𝐢𝐡. So we can write that two times the measure of angle 𝐴𝐢𝐡 is 136 degrees. By dividing through by two, we have that the measure of angle 𝐴𝐢𝐡 is 68 degrees. That means that both angles 𝐴𝐢𝐡 and 𝐴𝐡𝐢 have a measure of 68 degrees.

Now let’s consider the triangle 𝑀𝐡𝐢 and, in particular, the angle 𝑀𝐢𝐡. 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 𝑀𝐡𝐢 is 31 degrees, so triangle 𝑀𝐡𝐢 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 𝑀𝐡𝐢 is an isosceles triangle.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.