Question Video: Comparing the Lengths of Sides in a Triangle Mathematics

Video Transcript

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

In this question, we are given a figure containing information about various triangles. And we want to use this figure to compare two lengths in the figure: 𝐴𝐵 and 𝐶𝐷. To do this, we can start by noting that 𝐵𝐷 is the same length as 𝐶𝐷. If we then highlight 𝐴𝐵 on the diagram, we can see that 𝐴𝐵 and 𝐷𝐵 are both sides of triangle 𝐴𝐵𝐷. This means that we can compare the lengths of 𝐴𝐵 and 𝐶𝐷 by comparing the lengths of the sides in triangle 𝐴𝐵𝐷.

We can compare the side lengths in a triangle by using the side comparison theorem in triangles. This tells us that if one side in a triangle is opposite an angle of larger measure than another side, then it is longer. More formally, it tells us that if 𝑥𝑦𝑧 is a triangle and the measure of angle 𝑥 is greater than the measure of angle 𝑦, then 𝑦𝑧 is longer than 𝑥𝑧. Therefore, we can compare the lengths of these two sides by comparing the measures of the angles opposite them in triangle 𝐴𝐵𝐷. These are angles 𝐵𝐴𝐷 and 𝐴𝐷𝐵, as shown.

We can find the measure of angle 𝐴𝐷𝐵 by noting that it combines with angle 𝐵𝐷𝐶 to make a straight angle. So, their measures sum to give 180 degrees. Therefore, the measure of angle 𝐴𝐷𝐵 is equal to 180 degrees minus 92 degrees, which we can calculate is equal to 88 degrees. We can add this onto our diagram.

We now need to find the measure of angle 𝐵𝐴𝐷. To do this, we can note that sides 𝐶𝐷 and 𝐵𝐷 have the same length. So, triangle 𝐵𝐶𝐷 is an isosceles triangle. And we know that in an isosceles triangle, the angles opposite the congruent sides have equal measure. So, angles 𝐷𝐶𝐵 and 𝐷𝐵𝐶 have the same measure.

We can then use the fact that the sum of all of the measures of the internal angles in a triangle is 180 degrees to show that twice the measure of angle 𝐷𝐶𝐵 plus 92 degrees is equal to 180 degrees. We can then solve this equation for the measure of angle 𝐷𝐶𝐵. We subtract 92 degrees from both sides of the equation and divide through by two to get that the measure of angle 𝐷𝐶𝐵 is equal to 44 degrees. We can then add this angle measure onto the diagram to all of the angles congruent to angle 𝐷𝐶𝐵 as shown.

We now have two of the angle measures of triangle 𝐴𝐵𝐷. So, we can use the sum of the internal angle measures in a triangle to find the measure of the third angle. We have that the measure of angle 𝐵𝐴𝐷 is equal to 180 degrees minus 88 degrees minus 44 degrees. We can calculate that this is equal to 48 degrees. And we can add this onto the diagram as shown.

We can now apply the side comparison theorem in triangles to triangle 𝐴𝐵𝐷. We see that side 𝐴𝐵 is opposite an angle with larger measure than side 𝐷𝐵. So, 𝐴𝐵 must be longer than 𝐵𝐷. But remember, we are given in the diagram that 𝐵𝐷 and 𝐶𝐷 are the same length. So, we can also say that 𝐴𝐵 is greater than 𝐶𝐷, which is our final answer.