1. **State the problem:** We are given that the measure of an angle is forty-four times the measure of its supplementary angle. We need to find the measure of each angle. 2. **Recall the definition of supplementary angles:** Two angles are supplementary if the sum of their measures is 180 degrees. 3. **Set variables:** Let the measure of the smaller angle be $x$ degrees. 4. **Express the larger angle:** The larger angle is $44x$ degrees. 5. **Write the equation for supplementary angles:** $$x + 44x = 180$$ 6. **Simplify the equation:** $$45x = 180$$ 7. **Solve for $x$:** $$x = \frac{180}{45}$$ $$x = 4$$ 8. **Find the larger angle:** $$44x = 44 \times 4 = 176$$ 9. **Answer:** The two supplementary angles measure 4 degrees and 176 degrees respectively.