Question

Can a trapezoid and a parallelogram with the same base and height have the same area

Answer 1

No, unless you call a rectangle a trapezoid.

Recall the formula for the area of a parallelogram:
`bh`. Recall the area for a trapezoid:
`\frac{b+b_1}2h` (a trapezoid has two bases).

Suppose they did have the same area. Then
`bh=\frac{b+b_1}2h` and
`2b=\frac{b+b_1}` and
`b=b_1`.

So the only way for their areas to be equal is for the trapezoid to have the same base on top and bottom. That means the trapezoid must be a rectangle. Most people don't say a rectangle is a trapezoid, so no.