Video Transcript
Consider the figure shown. Fill in the blank with greater than, less than, or equal to: 𝐴𝐶 what 𝐶𝐷.
In this question, we are given a quadrilateral 𝐴𝐵𝐷𝐶 with a single pair of
parallel sides, so it is a trapezoid. We want to compare the lengths of two sides of this trapezoid by using the given
angle measures.
To answer this question, we can start by highlighting the two sides on the
diagram. It is worth noting that 𝐶𝐷 appears to be longer than 𝐴𝐶 in the figure. However, this is not a mathematically rigorous argument, so we should try to prove
that this is the case. We can compare the lengths of the two sides by noting that these are both side
lengths in triangle 𝐴𝐶𝐷. This means that we can compare the side lengths by comparing the measures of the
angles opposite the sides in triangle 𝐴𝐶𝐷.
We are given the measure of the angle opposite side 𝐶𝐷 in triangle 𝐴𝐶𝐷. This means that we need to find the measure of the angle opposite side 𝐴𝐶. We can do this by noting that the diagonal 𝐴𝐷 is a transversal of the parallel
lines in the trapezoid. We can then use the fact that alternate interior angles in a transversal of a pair of
parallel lines are congruent to note that the measure of angle 𝐴𝐷𝐶 is 31
degrees.
We can now compare the lengths of these two sides by using the side comparison
theorem in triangles. This tells us that if one side in a triangle is opposite an angle of larger measure
than another side in the triangle, then it must be longer. In our triangle 𝐴𝐶𝐷, we can see that the angle opposite side 𝐶𝐷 has greater
measure than the angle opposite 𝐴𝐶. So 𝐶𝐷 is longer than 𝐴𝐶. We can reverse this inequality to get that 𝐴𝐶 is less than 𝐶𝐷, giving us the
answer of less than.
