Video Transcript
Consider the figure shown. Fill in the blank with is greater
than, is less than, or is equal to: 𝐴𝐵 what 𝐴𝐶.
In this question, we are asked to
compare the lengths of two line segments on a figure. We can start by highlighting these
two line segments on the figure as shown. We can see that these are two of
the sides in triangle 𝐴𝐵𝐶. Since we want to compare side
lengths in a triangle, we can recall that the side comparison theorem in triangles
tells us that if the measure of one internal angle in a triangle has larger measure
than another internal angle, then the side opposite the angle with larger measure is
longer than the side opposite the angle with smaller measure.
Therefore, we can compare the
lengths of the sides in the figure by comparing the measures of the angles opposite
the sides in the triangle, that is, angles 𝐴𝐶𝐵 and 𝐴𝐵𝐶, respectively. We can determine the measure of the
angle at 𝐵 by noting that lines 𝐴𝐷 and 𝐵𝐶 are parallel. So it is a corresponding angle with
angle 𝐸𝐴𝐷. Hence, the measure of angle 𝐴𝐵𝐶
is 60 degrees.
In the same way, we can note that
the measure of the internal angle at 𝐶 must be 31 degrees. It is an alternate angle with angle
𝐶𝐴𝐷, so they are congruent. This means that we have shown that
the measure of the angle at 𝐵 is larger than the measure of the angle at 𝐶.
So the side comparison theorem in
triangles tells us that the side opposite the angle of larger measure is longer than
the side opposite the angle of smaller measure. Hence, line segment 𝐴𝐶 is longer
than line segment 𝐴𝐵. And we can say that 𝐴𝐵 is less
than 𝐴𝐶.
