Question Video: The Relation between Sides and Corresponding Angles in Triangles Mathematics

Video Transcript

In triangle 𝐴𝐵𝐶, the measure of angle 𝐴 is equal to 62 degrees and the measure of angle 𝐵 is equal to 92 degrees. Arrange the lengths of the sides of the triangle in descending order.

In this question, we are given the measures of two of the internal angles in a triangle and asked to use this to write the lengths of the sides of the triangle in descending order.

To answer this question, let’s start by sketching the information that we are given. We need to sketch a triangle 𝐴𝐵𝐶 with the angle at 𝐵 a little larger than a right angle and the angle at 𝐴 approximately 62 degrees. We get a sketch like the following.

In this sketch, it appears that 𝐴𝐶 is the longest side, followed by 𝐵𝐶, and then 𝐴𝐵 is the shortest side. However, we need to prove that this is the case for any triangle with these internal angle measures. We can do this by recalling that we can compare the side lengths in a triangle by comparing the measures of the angles opposite the sides by using the side comparison theorem in triangles. This tells us that if a side in a triangle is opposite an angle of larger measure than the angle opposite another side in that triangle, then it must be the longer of the two sides.

We can apply this result multiple times to note that if we write the angles in descending order of measure, then we can also write the sides in descending lengths. To apply this theorem, we need to know all the measures of the internal angles in the triangle. We can find the measure of angle 𝐶 by recalling that the sum of the internal angle measures in a triangle is 180 degrees. We can then find the measure of angle 𝐶 by calculating 180 degrees minus 92 degrees minus 62 degrees, which is equal to 26 degrees.

Now that we have the measure of all three internal angles of triangle 𝐴𝐵𝐶, we can see that the measure of angle 𝐵 is greater than the measure of angle 𝐴 is greater than the measure of angle 𝐶. Finally, we can use the side comparison theorem to conclude that the sides opposite each of the larger angles in turn will be longer. So 𝐴𝐶 is greater than 𝐵𝐶 is greater than 𝐴𝐵. We can write these lengths in descending order as 𝐴𝐶, 𝐵𝐶, 𝐴𝐵.