Question Video: Determining Whether a Given Triangle Is Isosceles Mathematics • Second Year of Preparatory School

Video Transcript

Is the triangle 𝐴𝐵𝐶 isosceles?

We can recall that an isosceles triangle is defined as a triangle that has two congruent sides. And because of the isosceles triangle theorem, we know that isosceles triangles also have two congruent angles. These base angles will be opposite the congruent sides. We can prove that a triangle is isosceles either by showing that a triangle has two congruent sides or by showing that it has two congruent angles. If we look at the figure, we have some information about the angles. So let’s see what we can work out.

Firstly, we have the straight line 𝐶𝐷, and we are given that the measure of angle 𝐴𝐵𝐷 is 142 degrees. We can work out the measure of angle 𝐴𝐵𝐶 by recalling that the angle measures on a straight line sum to 180 degrees. So the measure of angle 𝐴𝐵𝐶 is equal to 180 degrees minus 142 degrees, which is 38 degrees.

Now we have two angle measures in the triangle, we can work out the third angle by using the property that the internal angle measures in a triangle sum to 180 degrees. So, we have 38 degrees plus 83 degrees plus the measure of angle 𝐴𝐶𝐵 is equal to 180 degrees. Simplifying, we have 121 degrees plus the measure of angle 𝐴𝐶𝐵 is 180 degrees. And subtracting 121 degrees from both sides, we have that the measure of angle 𝐴𝐶𝐵 is 59 degrees. Therefore, this triangle has angles of 38 degrees, 59 degrees, and 83 degrees. There are no congruent angle measures.

So, we can give the answer to the question “Is the triangle 𝐴𝐵𝐶 isosceles?” as no, since we have proved that it does not have two congruent angles.