Opposite angles of a quadrilateral $ABCD$ are equal. If $AB = 4 \, cm$,determine $CD$ (in $cm$).

Prove that in a parallelogram,the bisectors of any two consecutive angles intersect at right angles.

$A$ square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

Which of the following is not true for a parallelogram?

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a parallelogram. If $AB = 12 \text{ cm}$ and $BC = 5 \text{ cm}$,then $AC = 13 \text{ cm}$.
$(2)$ $ABCD$ is a trapezium. If $AB \parallel CD$ and $AB = 10 \text{ cm}$,then $CD = 10 \text{ cm}$.

Diagonals of a parallelogram are perpendicular to each other. Is this statement true? Give reason for your answer.