Question Video: Applying the Side Comparison Theorem in Triangles to a Diagram Mathematics

Video Transcript

From the figure, fill in the blank with either greater than, less than, or equal to: 𝐴𝐶 what 𝐴𝐵.

In this question, we are given a figure containing the measures of some angles and a pair of parallel lines. We need to use this figure to compare the lengths of two line segments in the figure. To answer this question, let’s start by highlighting the two line segments whose lengths we want to compare. We can see that these line segments are both sides in triangle 𝐴𝐵𝐶.

We can compare the side lengths in a triangle by using the side comparison theorem in triangles. To do this, we need to find the measures of the angles opposite these two sides. We will do this by using the fact that 𝐴𝐶 and 𝐴𝐵 are transversals of a pair of parallel lines. First, we note that 𝐴𝐶 is a transversal of a pair of parallel lines. And we see that the angle at 𝐶 is the alternate interior angle of the given angle with measure 50 degrees.

Since alternate interior angles in a transversal of a pair parallel lines are congruent, we know that angle 𝐶 has measure 50 degrees. In the same way, we can see that 𝐴𝐵 is a transversal of the pair of parallel lines. And the angle at 𝐵 is the corresponding angle to the angle of measure 70 degrees. So these two angles are congruent.

We can now compare the lengths of these two sides by using the side comparison theorem in triangles which tells us that in a triangle, the side opposite the angle with larger measure will be longer. In this triangle, we can see that the angle opposite side 𝐴𝐶 has larger measure than the angle opposite side 𝐴𝐵. This means that 𝐴𝐶 is longer than 𝐴𝐵.

Hence, the answer to this question is greater than, since 𝐴𝐶 is greater than 𝐴𝐵 because the measure of angle 𝐵 is greater than the measure of angle 𝐶.