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

Video Transcript

Consider the given diagram. Fill in the blanks in the following statements using is greater than, is less than, or is equal to. 𝐴𝐶 what 𝐴𝐵. 𝐴𝐵 what 𝐵𝐶. 𝐷𝐶 what 𝐴𝐷. 𝐴𝐷 what 𝐴𝐶.

In this question, we are given a figure which contains the right triangle 𝐴𝐵𝐶. And we need to use this figure to compare the lengths of various line segments in the figure. To answer this question, we can begin by highlighting the first two line segments whose lengths we want to compare, 𝐴𝐶 and 𝐴𝐵. In the diagram, it appears as though 𝐴𝐶 is the longer side. However, we need to prove this. And we can do this by noting that both of these line segments are sides in triangle 𝐴𝐵𝐶.

We can then recall that we can compare the lengths of sides in a triangle by comparing the measures of the angles opposite the sides in the triangle by using the side comparison theorem in triangles. We can recall that 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 the side opposite vertex 𝑋 is longer than the side opposite vertex 𝑌. So side 𝑌𝑍 is longer than side 𝑋𝑍.

Therefore, we can compare the lengths of 𝐴𝐶 and 𝐴𝐵 by comparing the measures of the angles 𝐵 and 𝐶. We note that angle 𝐵 has a measure of 90 degrees, since it is a right angle. And the angle at 𝐶 has a measure of 45 degrees. Therefore, since 𝐴𝐶 is opposite an angle of larger measure than 𝐴𝐵, it must be the longer side. Hence, 𝐴𝐶 is greater than 𝐴𝐵.

This actually highlights a very useful application of the side comparison theorem in right triangles. We can note that the right angle is the angle of largest measure in any right triangle, since the sum of the three internal angles must be 180 degrees. This means that the side opposite the right angle must be the longest side in any right triangle. We call this the hypotenuse.

Let’s now move on to comparing the lengths of the line segments in the second part of the question. We can note that 𝐴𝐵 and 𝐵𝐶 are also sides in triangle 𝐴𝐵𝐶. We can once again try to use the side comparison theorem in triangles to compare the lengths of the two sides by comparing the measures of the angles opposite the two sides. We see that both sides are opposite angles of measure 45 degrees. This means that we can apply the isosceles triangle theorem to conclude that the sides opposite the angles of equal measure must have the same length. Hence, 𝐴𝐵 is equal to 𝐵𝐶.

Let’s now clear some space and move on to compare the lengths in the third part of the question. To do this, we can start by highlighting the two line segments whose lengths we want to compare as shown. We can see that these are both sides in triangle 𝐴𝐷𝐶. So we can try to compare their lengths by comparing the measures of the angles opposite the sides in this triangle. We can see that 𝐴𝐷 is opposite an angle of measure 45 degrees, whereas 𝐷𝐶 is opposite an angle of measure 25 degrees. So 𝐷𝐶 is opposite the angle of larger measure, so it is the longer side. Hence, 𝐷𝐶 is less than 𝐴𝐷.

Finally, we want to apply this process one more time to the last pair of sides. We can do this by noting that 𝐴𝐶 is the remaining side in triangle 𝐴𝐶𝐷. If we look at the angle opposite 𝐴𝐶, we can note that it is an obtuse angle. So we know it is the angle of largest measure in the triangle. So 𝐴𝐶 must be the longest side in the triangle. However, for due diligence, we can find the measure of this angle by using the fact that the sum of the internal angle measures in a triangle is 180 degrees to see that the measure of angle 𝐴𝐷𝐶 is equal to 180 degrees minus 45 degrees minus 25 degrees, which we can calculate is equal to 110 degrees. This confirms that 𝐴𝐶 is opposite the angle of larger measure, so 𝐴𝐶 is longer than 𝐴𝐷. And hence, 𝐴𝐷 is less than 𝐴𝐶.