$(1)$ The diagonals of a rhombus are ....... to each other.
$(2)$ The diagonals of a parallelogram ...... each other.

In a quadrilateral $ABCD$,$\angle A : \angle B : \angle C : \angle D = 4 : 5 : 4 : 5$. Find the measure of each angle of the quadrilateral and state the type of quadrilateral $ABCD$.

$A$ 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.

In the figure,$X$ and $Y$ are respectively the mid-points of the opposite sides $AD$ and $BC$ of a parallelogram $ABCD$. Also,$BX$ and $DY$ intersect $AC$ at $P$ and $Q$ respectively. Show that $AP = PQ = QC$.

$ABCD$ is a rhombus in which the altitude from $D$ to side $AB$ bisects $AB$. Find the angles of the rhombus.

$(1)$ $A$ quadrilateral has ......... pairs of consecutive angles.
$(2)$ The sum of measures of all the four angles of a quadrilateral is ...........