Video Transcript
Consider this triangle. Fill in the blanks in the following
statements with is equal to, is less than, or is greater than. The measure of angle 𝐴 what the
measure of angle 𝐵. The measure of angle 𝐵 what the
measure of angle 𝐶. The measure of angle 𝐶 what the
measure of angle 𝐴.
In this question, we are given a
triangle 𝐴𝐵𝐶 with known side lengths and asked to use this information to fill in
the blanks in three statements that compare the internal angle measures of this
triangle.
Since we want to compare the
measures of the angles in this triangle using the lengths of its sides, we can start
by recalling the angle comparison theorem in triangles, which says that the angles
opposite the longer sides have larger measure. More formally, if we have a
triangle 𝑋𝑌𝑍 and side 𝑥 is longer than side 𝑦, then the measure of angle 𝑋 is
larger than the measure of angle 𝑌. To apply this result to the given
triangle, we first need to compare the lengths of its sides. We note that the side opposite
vertex 𝐴 has length 15 centimeters. The side opposite vertex 𝐵 has
length 14 centimeters. And the side opposite vertex 𝐶 has
length 10 centimeters.
We can now compare the side
lengths. First, we note that side 𝑎 is
longer than side 𝑏. This means that side 𝑎 must be
opposite the angle of larger measure. So the measure of angle 𝐴 is
greater than the measure of angle 𝐵. We could follow the same process
with sides 𝑏 and 𝑐. However, it is worth noting that we
can compare all of the side lengths at the same time using compound
inequalities.
First, we note that 𝑐 is the
shortest side in the triangle. So we have that 𝑎 is longer than
𝑏 is longer than 𝑐. We can then apply the angle
comparison theorem in triangles to note that the angle at 𝐶 must have smaller
measure than the angles at 𝐴 and 𝐵. We can write this in the compound
inequality as shown. This then allows us to compare the
measures of any two angles in the triangle. We know that the angle at 𝐶 has
the smallest measure. So the measure of angle 𝐵 is
greater than the measure of angle 𝐶. And the measure of angle 𝐶 is less
than the measure of angle 𝐴.
