(a) Relationship between ABCD and EBCF:
Since ABCD and EBCF are on the same base BC and between the same parallel lines AF and BC, they are trapezoids. Specifically, ABCD is a trapezoid with bases AD and BC, while EBCF is a trapezoid with bases EF and BC. The trapezoids share the same height since they are between the same parallel lines AF and BC. Therefore, the relationship between ABCD and EBCF is that they are similar trapezoids.
(b) Proving the area of AFBC = area of ABCD:
Since ABCD and EBCF are similar trapezoids as shown in part (a), we know that the ratio of their areas is the square of the ratio of their corresponding sides.
Let's denote the height of both trapezoids as h.
The area of ABCD is given by:
\[ \text{Area of ABCD} = \frac{1}{2} (AD + BC) \times h \]
The area of EBCF is given by:
\[ \text{Area of EBCF} = \frac{1}{2} (EF + BC) \times h \]
Since ABCD and EBCF are similar, we have:
\[ \frac{\text{Area of ABCD}}{\text{Area of EBCF}} = \left(\frac{AD + BC}{EF + BC}\right)^2 \]
But EF = AD since ABCD and EBCF are on the same base BC. So, the ratio simplifies to:
\[ \frac{\text{Area of ABCD}}{\text{Area of EBCF}} = \left(\frac{AD + BC}{AD + BC}\right)^2 = 1 \]
This implies that the area of ABCD is equal to the area of EBCF. Since AFBC is part of ABCD, its area must be equal to the area of ABCD as well.
(c) Proving the area of quadrilateral AFEC = area of quadrilateral BCDF:
Since AB || FC || ED, we have AFEC and BCDF as parallelograms because opposite sides are parallel and equal in length. Therefore, the area of quadrilateral AFEC is equal to the area of quadrilateral BCDF.
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