1. **Stating the problem:** We need to find the area of the pentagon with given side lengths: two diagonal sides of 5 units each, two vertical sides of 4 units each, and a base of 8 units. 2. **Assumption:** To find the area, we assume the pentagon can be divided into simpler shapes such as rectangles and triangles. 3. **Step 1:** Consider the pentagon as a rectangle of width 8 units and height 4 units, plus two right triangles on the sides with legs 4 units (height) and 3 units (half the difference between 8 and the base of the triangles). 4. **Calculate the base of each triangle:** $$\text{base} = \frac{8 - (5 + 5 - 8)}{2} = \frac{8 - 2}{2} = 3$$ 5. **Calculate the area of the rectangle:** $$A_{rectangle} = 8 \times 4 = 32$$ 6. **Calculate the area of one triangle:** $$A_{triangle} = \frac{1}{2} \times 3 \times 4 = 6$$ 7. **Calculate the total area of two triangles:** $$2 \times 6 = 12$$ 8. **Calculate the total area of the pentagon:** $$A = A_{rectangle} + 2 \times A_{triangle} = 32 + 12 = 44$$ 9. **Answer:** The area of the pentagon is $44$ square units.