To solve for the area of the pentagon, we can break down the shape into smaller, manageable geometric shapes—specifically triangles and rectangles. The provided information includes side lengths and angles.

Step-by-Step Solution:

1. Identify shapes:

   • The pentagon can be divided into a right triangle (bottom-left corner) and two non-right triangles.

2. Right Triangle Area:

   • The right triangle on the left side has two sides of length 1, so the area is: $\text{Area}_{\text{right triangle}} = \frac{1}{2} \times 1 \times 1 = 0.5$

3. Upper Triangle Area (Angle = 60°):

   • For the triangle formed by sides of lengths 1 and 1 with a 60° angle between them, we use the formula for the area of a triangle with known sides and included angle: $\text{Area} = \frac{1}{2} \times a \times b \times \sin(\theta)$ where $a = 1$, $b = 1$, and $\theta = 60^\circ$. $\text{Area}_{60^\circ \text{ triangle}} = \frac{1}{2} \times 1 \times 1 \times \sin(60^\circ) = \frac{\sqrt{3}}{4} \approx 0.433$

4. Bottom Triangle Area (Angle = 150°):

   • For the triangle with side lengths 1 and 2 and a 150° angle, using the same formula: $\text{Area}_{150^\circ \text{ triangle}} = \frac{1}{2} \times 1 \times 2 \times \sin(150^\circ) = \frac{1}{2} \times 1 \times 2 \times \frac{1}{2} = 0.5$

5. Sum the Areas:

   • Add up the areas of the three parts: $\text{Total Area} = 0.5 (\text{right triangle}) + 0.433 (\text{60° triangle}) + 0.5 (\text{150° triangle}) = 1.433$

Thus, the total area of the pentagon is approximately 1.433 square units.

Would you like a more detailed explanation of any step? Here are some related questions:

1. How do you find the area of a triangle using side lengths and angles?
2. What is the sine of common angles like 30°, 60°, and 90°?
3. How do you break complex polygons into simpler shapes for area calculation?
4. How does the Law of Sines help in solving non-right triangles?
5. Can we calculate the perimeter of this pentagon with the given sides?

Tip: For complex polygons, divide them into smaller shapes like triangles or rectangles to simplify area calculations.