1. **Problem statement:** We have a polygon divided into two parallelograms with given angles 48° and 117° at the top parts. We need to find the angle measure at the bottom point. 2. **Key properties:** In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to 180°). 3. **Step 1:** Identify the angles adjacent to the bottom angle. Since the polygon is formed by two parallelograms, the angles 48° and 117° are at the top, and their adjacent angles inside the parallelograms are supplementary. 4. **Step 2:** Calculate the angles adjacent to 48° and 117° inside the parallelograms: $$180^\circ - 48^\circ = 132^\circ$$ $$180^\circ - 117^\circ = 63^\circ$$ 5. **Step 3:** The bottom angle is formed by these two adjacent angles (132° and 63°) meeting at the bottom point. Since the figure is concave, the bottom angle is the difference between these two angles: $$132^\circ - 63^\circ = 69^\circ$$ 6. **Step 4:** However, the bottom angle is the external angle to the concave vertex, so the actual angle measure is: $$180^\circ - 69^\circ = 111^\circ$$ 7. **Answer:** The angle measure at the bottom point is 111.