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

Is ππ΅πΆ an isosceles triangle?

03:18

Video Transcript

Is ππ΅πΆ an isosceles triangle?

Letβ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 π΄πΆ and π΄π΅ are congruent. But be careful, because this tells us that triangle π΄π΅πΆ is an isosceles triangle. And we need to determine if triangle ππ΅πΆ is isosceles. So letβ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 π΄π΅πΆ 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 π΄π΅πΆ, these base angles will be the angles π΄πΆπ΅ and π΄π΅πΆ. Letβ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 π΄π΅πΆ β thatβs the measure of angle π΄πΆπ΅, the measure of angle π΄π΅πΆ, and the measure of angle πΆπ΄π΅ β will all sum to 180 degrees. And we were given that the measure of angle πΆπ΄π΅ 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 weβve already noted that these two angles are congruent, that is, that the measure of angle π΄π΅πΆ is equal to the measure of angle π΄πΆπ΅. So we can write that two times the measure of angle π΄πΆπ΅ is 136 degrees. By dividing through by two, we have that the measure of angle π΄πΆπ΅ is 68 degrees. That means that both angles π΄πΆπ΅ and π΄π΅πΆ have a measure of 68 degrees.

Now letβs consider the triangle ππ΅πΆ and, in particular, the angle ππΆπ΅. 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 ππ΅πΆ is 31 degrees, so triangle ππ΅πΆ 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 ππ΅πΆ is an isosceles triangle.