Question: The measure of an angle is thirty-nine times the measure of its supplementary angle. What is the measure of each angle?
1. **State the problem:** We need to find two supplementary angles where one angle is thirty-nine times the other.
2. **Recall the definition of supplementary angles:** Two angles are supplementary if their measures add up to $$180^\circ$$.
3. **Set variables:** Let the measure of the smaller angle be $$x$$ degrees.
4. **Express the larger angle:** The larger angle is $$39x$$ degrees.
5. **Write the equation for supplementary angles:**
$$x + 39x = 180$$
6. **Simplify the equation:**
$$40x = 180$$
7. **Solve for $$x$$:**
$$x = \frac{180}{40}$$
8. **Show cancellation:**
$$x = \frac{\cancel{180}}{\cancel{40}} = \frac{9}{2} = 4.5$$
9. **Find the larger angle:**
$$39x = 39 \times 4.5 = 175.5$$
10. **Answer:** The two angles measure $$4.5^\circ$$ and $$175.5^\circ$$ respectively.