can a trapezoid and a parallelogram with the same base and height have the same area ?

No, a trapezoid has both sides angling inward, while the parallelogram only has one going inward (toward the top).

the two bases of a parallelogram are equal

the two bases of a trapezoid are different

Bh vs (b+B)h/2

These can only be the same if b=B.

Yes, a trapezoid and a parallelogram with the same base and height can indeed have the same area. To understand why, let's first define a trapezoid and a parallelogram.

A trapezoid is a quadrilateral with at least one pair of parallel sides. It has two bases, the top base and the bottom base, which are parallel to each other. The height of a trapezoid is the perpendicular distance between the two bases.

On the other hand, a parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. The height of a parallelogram is the perpendicular distance between the two parallel sides.

To determine the area of a trapezoid, you can use the formula:

Area = ((base1 + base2) * height) / 2

Where base1 is the longer base, base2 is the shorter base, and height is the perpendicular distance between the bases.

For a parallelogram, the formula for calculating the area is:

Area = base * height

Note that both formulas involve multiplying the base by the height. If a trapezoid and parallelogram have the same base and height, the product of the base and the height will be the same. Therefore, they will have the same area.

So, the answer to your question is yes, a trapezoid and a parallelogram with the same base and height can have the same area.