### Video Transcript

A pair of supplementary angles are
in the ratio of one to nine. What is the measure of the smaller
angle?

In this question, we are told that
a pair of supplementary angles are in the ratio of one to nine. This means that one angle is nine
times the size of the other. We need to use this information to
determine the measure of the smaller angle.

To answer this question, we can
first recall that we call two angles π΄ and π΅ supplementary if their measures sum
to 180 degrees. We can note that the angles are in
a ratio of one to nine. If we say that angle π΄ is the
angle with smaller measure, then this means that nine times the measure of angle π΄
is equal to the measure of angle π΅.

We want to find the measure of
angle π΄ since it is the smaller angle. We can do this by substituting this
expression of the measure of angle π΄ into the equation of the measure of the
supplementary angles. This gives us that the measure of
angle π΄ plus nine times the measure of angle π΄ is equal to 180 degrees. We can then simplify the left-hand
side of the equation to get 10 times the measure of angle π΄. We can then solve for the measure
of the smaller angle by dividing the equation through by 10. We get that the smaller angle has
measure 18 degrees.

We can check this answer by
multiplying the measure by nine to find that the measure of the larger angle is 162
degrees. We can check that these angles are
indeed supplementary by checking that the sum of their measures is 180 degrees. We can calculate that 18 degrees
plus 162 degrees is 180 degrees, confirming that the angles are supplementary.

Hence, if two supplementary angles
are in the ratio of one to nine, we have shown that the smaller angle must have
measure 18 degrees.