Answer:
To prove that Parallelogram ABCD is equal to the sum of Parallelogram ADEF and Parallelogram BCEF, we need to use the property of area addition of parallelograms.
Given that ABCD, ADEF, and BCEF are parallelograms, we know that the opposite sides are parallel and equal in length.
Let's denote the areas of the parallelograms as follows:
Area of Parallelogram ABCD = [ABCD]
Area of Parallelogram ADEF = [ADEF]
Area of Parallelogram BCEF = [BCEF]
Now, we will prove that [ABCD] = [ADEF] + [BCEF]:
1. Since ABCD and ADEF are parallelograms, they share a common base AD and are on the same base AD and between the same parallels AD and BC. Therefore, the area of Parallelogram ABCD = base AD height h1 and the area of Parallelogram ADEF = base AD height h2. This gives us:
[ABCD] = AD * h1
[ADEF] = AD * h2
2. Similarly, since ABCD and BCEF are parallelograms, they share a common base BC and are on the same base BC and between the same parallels AD and BC. Therefore, the area of Parallelogram ABCD = base BC height h1 and the area of Parallelogram BCEF = base BC height h3. This gives us:
[ABCD] = BC * h1
[BCEF] = BC * h3
3. Now, we can rewrite the areas of the parallelograms in terms of the heights h1, h2, and h3:
[ABCD] = AD h1 = BC h1
[ADEF] = AD * h2
[BCEF] = BC * h3
4. Adding the areas of ADEF and BCEF:
[ADEF] + [BCEF] = AD h2 + BC h3
[ABCD] = AD h1 = BC h1
5. Since h1 = h2 + h3 (as the heights together form the height of ABCD), we can substitute this into the equation:
[ABCD] = AD (h2 + h3) = AD h2 + AD * h3
6. By rearranging terms, we get:
[ABCD] = [ADEF] + [BCEF]
Therefore, we have proved that Parallelogram ABCD is equal to the sum of Parallelogram ADEF and Parallelogram BCEF
