Video Transcript
Consider the shown triangle
𝐴𝐵𝐶. For each of the following
statements, fill in the blank with is greater than, is less than, or is equal
to. 𝐴𝐶 what 𝐴𝐵, 𝐴𝐵 what 𝐵𝐷,
𝐶𝐷 what 𝐴𝐷, and 𝐵𝐷 what 𝐴𝐶.
In this question, we are given a
figure of a triangle. And we want to use this figure to
compare the lengths of various line segments in the figure. To do this, let’s start by
highlighting the first pair of line segments whose lengths we want to compare: 𝐴𝐶
and 𝐴𝐵. We can then note that these are
both sides in triangle 𝐴𝐵𝐶. So we can compare the lengths of
the sides by recalling the side comparison theorem in triangles. This tells us that if one side of a
triangle is opposite an angle of larger measure, then it must be the longer of the
two sides. More formally, this tells us that
if we have a triangle 𝑋𝑌𝑍 and the measure of angle 𝑋 is greater than the measure
of angle 𝑌, then 𝑌𝑍 is longer than 𝑋𝑍 since it is opposite an angle of larger
measure in the triangle.
Therefore, we can compare the
lengths of these two sides by comparing the measures of the angles opposite them in
triangle 𝐴𝐵𝐶. We see that 𝐴𝐶 is opposite the
right angle of measure 90 degrees and 𝐴𝐵 is opposite an angle of measure 30
degrees. Hence, since 𝐴𝐶 is opposite an
angle of larger measure than 𝐴𝐵, it must be the longer side. So 𝐴𝐶 is greater than 𝐴𝐵.
It is worth noting that the angle
comparison theorem tells us that the side opposite the right angle in any right
triangle is always the longest, since the right angle is the angle with the largest
measure. We call this side the
hypotenuse. We can then use this idea to answer
the fourth part of this question. 𝐴𝐶 is the longest side in
triangle 𝐴𝐵𝐶, so it is longer than 𝐵𝐶, which in turn is longer than 𝐵𝐷. So, we know that 𝐵𝐷 is less than
𝐴𝐶.
Let’s now try to apply the side
comparison theorem to the sides in the second part of the question. First, we highlight the two sides
whose lengths we want to compare as shown. We see that these are both side
lengths in triangle 𝐴𝐵𝐷. We can use the side comparison
theorem in triangles to compare the lengths of these sides by comparing the measure
of the angles opposite the sides in triangle 𝐴𝐵𝐷. We see that angle 𝐵𝐴𝐷 has
measure 30 degrees. And we can find the measure of
angle 𝐴𝐷𝐵 by recalling that the sum of the internal angle measures in a triangle
is 180 degrees. Therefore, the measure of angle
𝐴𝐷𝐵 is equal to 180 degrees minus 90 degrees minus 30 degrees, which is equal to
60 degrees. We can then see that 𝐴𝐵 is
opposite an angle of larger measure, so it must be the longer of the two sides. Hence, 𝐴𝐵 is greater than
𝐵𝐷.
Let’s apply this process one final
time to the pair of sides in the third part of this question. We can start by highlighting the
two sides whose lengths we want to compare on the figure. We see that these are both side
lengths in triangle 𝐴𝐶𝐷. If we look at the angle measures
opposite the sides in triangle 𝐴𝐶𝐷, we see that both sides are opposite angles of
measure 30 degrees. Since the sides are opposite angles
of equal measure, we can use the isosceles triangle theorem to conclude that the
sides have the same length. So, 𝐶𝐷 is equal to 𝐴𝐷.
