Diagonals of a rectangle PQRS are intersecting in point M. If $\angle QMR=50°$ then the measure of $\angle MPS$ is $k°$. Find the value of $k$.

In the given figure, if

seg $AB \parallel$ seg $PQ$, seg $AB \cong$ seg $PQ$,

seg $AC \parallel$ seg $PR$, seg $AC\cong$ seg $PR$,

then prove that,

seg $BC \parallel$ seg $QR$ and seg $BC\cong$ seg $QR$.

In the given figure, $ABCD$ is a trapezium. $AB\parallel DC$. Points $P$ and $Q$ are midpoints of seg $AD$ and seg $BC$ respectively.

Then prove that, $PQ\parallel AB$ and $PQ=\frac{1}{2}\left(AB+DC\right)$.

In the adjacent figure, $\square ABCD$ is a trapezium. $AB\parallel DC$. Points $M$ and $N$ are midpoints of diagonal $AC$ and $DB$ respectively then prove that $MN\parallel AB$.