Question Video: Comparing Side Lengths in a Triangle Using Angle Measures Mathematics • 11th Grade

Video Transcript

Consider the shown triangle 𝐴𝐵𝐶. For each of the following statements, fill in the blank with is greater than, is less than, or is equal to. 𝐴𝐶 what 𝐴𝐵, 𝐴𝐵 what 𝐵𝐷, 𝐶𝐷 what 𝐴𝐷, and 𝐵𝐷 what 𝐴𝐶.

In this question, we are given a figure of a triangle. And we want to use this figure to compare the lengths of various line segments in the figure. To do this, let’s start by highlighting the first pair of line segments whose lengths we want to compare: 𝐴𝐶 and 𝐴𝐵. We can then note that these are both sides in triangle 𝐴𝐵𝐶. So we can compare the lengths of the sides by recalling the side comparison theorem in triangles. This tells us that if one side of a triangle is opposite an angle of larger measure, then it must be the longer of the two sides. More formally, this tells us that if we have a triangle 𝑋𝑌𝑍 and the measure of angle 𝑋 is greater than the measure of angle 𝑌, then 𝑌𝑍 is longer than 𝑋𝑍 since it is opposite an angle of larger measure in the triangle.

Therefore, we can compare the lengths of these two sides by comparing the measures of the angles opposite them in triangle 𝐴𝐵𝐶. We see that 𝐴𝐶 is opposite the right angle of measure 90 degrees and 𝐴𝐵 is opposite an angle of measure 30 degrees. Hence, since 𝐴𝐶 is opposite an angle of larger measure than 𝐴𝐵, it must be the longer side. So 𝐴𝐶 is greater than 𝐴𝐵.

It is worth noting that the angle comparison theorem tells us that the side opposite the right angle in any right triangle is always the longest, since the right angle is the angle with the largest measure. We call this side the hypotenuse. We can then use this idea to answer the fourth part of this question. 𝐴𝐶 is the longest side in triangle 𝐴𝐵𝐶, so it is longer than 𝐵𝐶, which in turn is longer than 𝐵𝐷. So, we know that 𝐵𝐷 is less than 𝐴𝐶.

Let’s now try to apply the side comparison theorem to the sides in the second part of the question. First, we highlight the two sides whose lengths we want to compare as shown. We see that these are both side lengths in triangle 𝐴𝐵𝐷. We can use the side comparison theorem in triangles to compare the lengths of these sides by comparing the measure of the angles opposite the sides in triangle 𝐴𝐵𝐷. We see that angle 𝐵𝐴𝐷 has measure 30 degrees. And we can find the measure of angle 𝐴𝐷𝐵 by recalling that the sum of the internal angle measures in a triangle is 180 degrees. Therefore, the measure of angle 𝐴𝐷𝐵 is equal to 180 degrees minus 90 degrees minus 30 degrees, which is equal to 60 degrees. We can then see that 𝐴𝐵 is opposite an angle of larger measure, so it must be the longer of the two sides. Hence, 𝐴𝐵 is greater than 𝐵𝐷.

Let’s apply this process one final time to the pair of sides in the third part of this question. We can start by highlighting the two sides whose lengths we want to compare on the figure. We see that these are both side lengths in triangle 𝐴𝐶𝐷. If we look at the angle measures opposite the sides in triangle 𝐴𝐶𝐷, we see that both sides are opposite angles of measure 30 degrees. Since the sides are opposite angles of equal measure, we can use the isosceles triangle theorem to conclude that the sides have the same length. So, 𝐶𝐷 is equal to 𝐴𝐷.