Video Transcript
Arrange the side lengths of triangle 𝐴𝐶𝐷 in descending order.
In this question, we are given a figure and we need to use this figure to compare the side lengths in a triangle. Since this figure contains angle measures, we can begin by recalling the side comparison theorem in triangles. We can recall that this tells us 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. In other words, we can compare side lengths in a triangle by comparing the measures of the angles opposite the sides.
We can also recall that we can extend this to include all three sides in the triangle. Hence, we can arrange the side lengths of triangle 𝐴𝐶𝐷 into descending order by rearranging its angle measures into descending order. To do this, we will find its internal angle measures.
In the figure, we can see that lines 𝐴𝐵 and 𝐶𝐷 are parallel. So, the line passing through 𝐴 and 𝐶 is a transversal of parallel lines. We can use this to note that angle 𝐴𝐶𝐷 is an alternate angle with angle 𝐵𝐴𝐶. So, they are congruent. Hence, the measure of angle 𝐴𝐶𝐷 is 55 degrees.
We can find the measure of angle 𝐴𝐷𝐶 by recalling that the sum of the internal angle measures in a triangle is 180 degrees. Therefore, the measure of angle 𝐴𝐷𝐶 is equal to 180 degrees minus 55 degrees minus 52 degrees, which we can calculate is equal to 73 degrees.
Now that we have found all of the measures of the internal angles in triangle 𝐴𝐶𝐷, we can arrange the measures in descending order. We have that the measure of the angle at 𝐷 is greater than the measure of the angle at 𝐶, which is greater than the measure of the angle at 𝐴, since 73 degrees is greater than 55 degrees, which is greater than 52 degrees.
Now, we can apply the side comparison theorem in triangles to conclude that the sides opposite the angles with larger measure are longer than the sides opposite the angles with smaller measure, to find that 𝐴𝐶 is greater than 𝐴𝐷, which is greater than 𝐶𝐷. Hence, 𝐴𝐶, 𝐴𝐷, 𝐶𝐷 is the side lengths of triangle 𝐴𝐶𝐷 written in descending order.
