Let's denote the two supplementary angles as $x$ and $y$.

Given that the angles are supplementary, we have: $x + y = 180^\circ$ We are also told that one angle is one-third the measure of the other. Let's assume $x$ is the smaller angle, so: $x = \frac{1}{3}y$

Now we can substitute $x = \frac{1}{3}y$ into the equation $x + y = 180^\circ$: $\frac{1}{3}y + y = 180^\circ$

To solve for $y$, first combine the terms on the left-hand side: $\frac{1}{3}y + \frac{3}{3}y = 180^\circ$ $\frac{4}{3}y = 180^\circ$

Next, solve for $y$ by multiplying both sides by $\frac{3}{4}$: $y = 180^\circ \times \frac{3}{4}$ $y = 135^\circ$

Since $x = \frac{1}{3}y$, substitute $y = 135^\circ$ into this equation to find $x$: $x = \frac{1}{3} \times 135^\circ = 45^\circ$

Thus, the two supplementary angles are $45^\circ$ and $135^\circ$.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:

1. What are complementary angles, and how do they differ from supplementary angles?
2. If one angle is half the measure of its supplementary angle, what are the two angles?
3. Can two obtuse angles be supplementary? Why or why not?
4. What is the relationship between the two angles if they are both equal and supplementary?
5. How would the problem change if the angles were complementary instead of supplementary?

Tip: Remember, supplementary angles always add up to $180^\circ$, while complementary angles add up to $90^\circ$.