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

Consider the given diagram. Fill in the blanks in the following statements using >, <, or =. π΄πΆ οΌΏ π΄π΅. π΄π΅ οΌΏ π΅πΆ. π·πΆ οΌΏ π΄π·. π΄π· οΌΏ π΄πΆ.

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### 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. π΄πΆ what π΄π΅. π΄π΅ what π΅πΆ. π·πΆ what π΄π·. π΄π· what π΄πΆ.

In this question, we are given a figure which contains the right triangle π΄π΅πΆ. 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, π΄πΆ and π΄π΅. In the diagram, it appears as though π΄πΆ 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 π΄π΅πΆ.

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 πππ and the measure of angle π is greater than the measure of angle π, then the side opposite vertex π is longer than the side opposite vertex π. So side ππ is longer than side ππ.

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

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.

Letβs now move on to comparing the lengths of the line segments in the second part of the question. We can note that π΄π΅ and π΅πΆ are also sides in triangle π΄π΅πΆ. 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, π΄π΅ is equal to π΅πΆ.

Letβ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 π΄π·πΆ. 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 π΄π· is opposite an angle of measure 45 degrees, whereas π·πΆ is opposite an angle of measure 25 degrees. So π·πΆ is opposite the angle of larger measure, so it is the longer side. Hence, π·πΆ is less than π΄π·.

Finally, we want to apply this process one more time to the last pair of sides. We can do this by noting that π΄πΆ is the remaining side in triangle π΄πΆπ·. If we look at the angle opposite π΄πΆ, we can note that it is an obtuse angle. So we know it is the angle of largest measure in the triangle. So π΄πΆ 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 π΄π·πΆ is equal to 180 degrees minus 45 degrees minus 25 degrees, which we can calculate is equal to 110 degrees. This confirms that π΄πΆ is opposite the angle of larger measure, so π΄πΆ is longer than π΄π·. And hence, π΄π· is less than π΄πΆ.