Question Video: Finding the Measure of an Angle given a Relation with Its Supplementary Angle’s Measure Mathematics

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

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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 𝐴 and 𝐡 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 𝐴 is the angle with smaller measure, then this means that nine times the measure of angle 𝐴 is equal to the measure of angle 𝐡.

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

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