Prove that in a rectangle, the diagonals are of equal lengths.

In the figure below, $ABCD$ is a parallelogram. $M$ is the midpoint of $AC$; $X, Y$ are points on $AB$ and $DC$ respectively such that $AX=CY$. Prove that: $△AXM\cong △CYM$ 

In the figure below, $ABCD$ is a parallelogram. $M$ is the midpoint of $AC$; $X, Y$ are points on $AB$ and $DC$ respectively such that $AX=CY$. Prove that: $XMY$ is a straight line.

Prove that: A diagonal of a square makes an angle of $45°$ with the side of the square.

Prove that: The diagonals of a rhombus are at right angles.

Prove that: The diagonals of a rhombus bisects the angles at vertices.

In the figure below, $PQRS$ is a parallelogram. $QU$ and $ST$ are perpendiculars on the diagonal $PR$. Prove that: $△STR\cong △QUP$

In the figure below, $PQRS$ is a parallelogram. $QU$ and $ST$ are perpendiculars on the diagonal $PR$. Prove that: $ST=QU$.

In figure below $PQRS$ is a parallelogram. $PO$ and $QO$ are respectively the angle bisectors of $\angle P$ and $\angle Q$. Line $LOM$ is drawn parallel to $PQ$. Prove that: $PL=QM$