In an affine plane are given a trapezoid $T$ and a parallelogram $P$. Can exist an affinity $f$ which maps $T$ in $P$? Justify the answer.
I know that parallelism is an affine property, which means that parallel segments are mapped into parallel segments. Hence, to exist such an affinity $f$, it must map the pair of parallel sides of $T$ into one of the pair of the parallel sides of $P$. My questions now are: what about the other pair of nonparallel sides of $T$? If they are nonparallel, does it mean that their images under $f$ are necessarly nonparallel, or can they be parallel? And what about the internal points of $T$? Must their images under $f$ be also internal points of $P$?
The inverse of an affine transformation is another affine transformation. If you transform the trapezoid $T$ into a parallelogram $P$, the inverse transformation must preserve parallel lines, so it must map the parallel pairs of sides of $P$ to parallel pairs of sides of $T$. Thus, $T$ must have already been a parallelogram.
(The exception is non-invertible affine transformations, in which the entire plane is mapped to a single line. Here, it's pretty clear that non-parallel lines can become parallel; but that's not what's going on in your case.)
As for internal points, this follows because affine transformations map line segments to line segments. A point on the line segment $[\mathbf x, \mathbf y]$ has the form $t \mathbf x + (1-t)\mathbf y$ for some $t \in [0,1]$; if $f$ is an affine transformation, then we have $f(t \mathbf x + (1-t)\mathbf y) = t f(\mathbf x) + (1-t) f(\mathbf y)$, a point on the line segment $[f(\mathbf x), f(\mathbf y)]$.
Consequently, if $f$ maps the points $A, B, C, D$ (vertices of $T$) to $f(A), f(B), f(C), f(D)$ (vertices of $P$), then by one application of the property above it must map the sides $AB, BC, CD, AD$ of $T$ to the sides of $P$, and by another application it must map the convex hull of $\{A,B,C,D\}$ to the convex hull of $\{f(A), f(B), f(C), f(D)\}$.