### Video Transcript

Arrange the side lengths of triangle 𝐴𝐵𝐶 in ascending order.

In this question, we are given a triangle and the measure of two internal angles of the triangle. And we are asked to use this to arrange the sides of the triangle in ascending order of length. Since we are asked to compare the lengths of the sides in the triangle, we can begin by recalling the side comparison theorem in triangles. This tells us that in a triangle 𝐴𝐵𝐶, if one side is opposite an angle of larger measure than another side, then it is the longer of the two sides. This means that we can compare the side lengths in a triangle by comparing the measures of the angles opposite them.

We can find the measure of angle 𝐴 by recalling that the sum of the internal angle measures in a triangle is 180 degrees. So we have the measure of angle 𝐴 plus 59 degree plus 94 degrees is equal to 180 degrees. We can solve this equation for the measure of angle 𝐴. We have that the measure of angle 𝐴 is 180 degrees minus 94 degrees minus 59 degrees, which we can calculate is equal to 27 degrees.

We can now arrange the angle measures in the triangle into descending order of measure. We have that the measure of angle 𝐵 is greater than the measure of angle 𝐶 is greater than the measure of angle 𝐴. We can then use the side comparison theorem to note that the side opposite each angle in turn must be longer than the last. So 𝐴𝐶 is greater than 𝐴𝐵 is greater than 𝐵𝐶. We can now reverse the orders of these sides to write them in ascending order. We have that line segment 𝐵𝐶 is the shortest, followed by line segment 𝐴𝐵, and finally line segment 𝐴𝐶 is the longest.