M L Aggarwal Solutions for Exercise 1: EXERCISE 13.1

In the figure given below, $ABCD$ is a rhombus. Find the value of $x$ (in degrees).

In the figure given below, equilateral triangle $EBC$ surmounts square $ABCD$. Find angle $BED$ represented by $x$ (in degrees).

The angles of a quadrilateral are equal. Prove that the quadrilateral is a rectangle.

$\mathrm{ABCD}$ is a rectangle in which diagonal $BD$ bisects $\angle B$. Show that $ABCD$ is a square.

Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

$P$ and $Q$ are points on opposite sides $\mathrm{AD}$ and $\mathrm{BC}$ of a parallelogram $\mathrm{ABCD}$ such that $\mathrm{PQ}$ passes through the point of intersection $\mathrm{O}$ of its diagonals $\mathrm{AC}$ and $\mathrm{BD}$. Show that $\mathrm{PQ}$ is bisected at $\mathrm{O}$.

$ABCD$ is a square. $A$ is joined to a point $P$ on $BC$ and $D$ is joined to a point $Q$ on $AB$. If $AP=DQ$, prove that $AP$ and $DQ$ are perpendicular to each other.

In a parallelogram $ABCD$, the bisector of $\angle A$ meets $DC$ at $E$ and $AB=2AD$. Prove that $BE$ bisects $\angle B$.