Show that each angle of a rectangle is a right angle.

$ABC$ is an isosceles triangle in which $AB=AC$. $AD$ bisects exterior angle $PAC$ and $CD\parallel AB$ (see Fig.). Show that $\angle DAC=\angle BCA$.

$ABC$ is an isosceles triangle in which $AB=AC$. $AD$ bisects exterior angle $PAC$ and $CD\parallel AB$ (see Fig.). Show that $ABCD$ is a parallelogram.

Two parallel lines $l$ and $m$ are intersected by a transversal $p$ (see Fig.). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.   

$ABCD$ is a parallelogram in which $P$ and $Q$ are mid-points of opposite sides $AB$ and $CD$ (see Fig.). If $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$, show that $APCQ$ is a parallelogram, $DPBQ$  is a parallelogram and $PSQR$ is a parallelogram.

In $△ABC, D, E$ and $F$ are respectively the mid-points of sides $AB, BC$ and $CA$(see Fig.). Show that $△ABC$ is divided into four congruent triangles by joining $D, E$ and $F$.

$l, m$ and $n$ are three parallel lines intersected by transversals $p$ and $q$ such that $l, m$ and $n$ cut off equal intercepts $AB$ and $BC$ on $p$ (see Fig. ). Show that $l, m$ and $n$ cut off equal intercepts $DE$ and $EF$ on $q$ also.