$ABCD$ is a square. $ABO$ is an equilateral triangle inside the square. Find $\angle DOC.$

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: $△STR\cong △QUP$