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.
