Question Video: Applying the Side Comparison Theorem and Isosceles Triangle Theorem Mathematics • 11th Grade

Video Transcript

Consider the figure shown. Fill in the blank with is greater than, is less than, or is equal to. 𝐴𝐵 what 𝐵𝐶.

In this question, we are asked to compare the lengths of two sides in a given figure. If we highlight the two sides whose lengths we want to compare, then we can see that they are both side lengths in triangle 𝐴𝐵𝐶. We can then recall that we can compare the lengths of sides in a triangle by using the side comparison theorem for triangles, which tells us that if one side of a triangle is opposite an angle of larger measure than another side, then it must be longer. More formally, we have that if 𝑥𝑦𝑧 is a triangle with the measure of angle 𝑥 greater than the measure of angle 𝑦, then 𝑦𝑧 must be longer than 𝑥𝑧.

To apply this theorem to triangle 𝐴𝐵𝐶, we need to compare the measures of the angles opposite the two highlighted sides. This means that we want to compare the measures of the angles at 𝐴 and 𝐶. We can do this by noting that triangle 𝐴𝐶𝐷 has a pair of congruent sides. So, it is an isosceles triangle. We can then recall that the angles opposite the congruent sides in an isosceles triangle have the same measure. So, the measure of angle 𝐷𝐴𝐶 is equal to the measure of angle 𝐶. We can then see that the angle at 𝐴 has measure equal to the measure of angle 𝐷𝐴𝐶 plus the measure of angle 𝐵𝐴𝐷.

Since the measure of angle 𝐵𝐴𝐷 must be positive since it is not the zero angle and the measure of angle 𝐷𝐴𝐶 is equal to the measure of angle 𝐶, we can note that the measure of angle 𝐴 must be greater than the measure of angle 𝐶. Therefore, by the side comparison theorem in triangles, the side opposite angle 𝐴 is longer than the side opposite angle 𝐶. Hence, 𝐵𝐶 is longer than 𝐴𝐵. And we can say that 𝐴𝐵 is less than 𝐵𝐶.