If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Show that the diagonals of a square are equal and bisect each other at right angles.
Diagonal \(AC\) of a parallelogram \(ABCD\) bisects \(\angle A\). Show that:
(i) it bisects \(\angle C\) also,
(ii) \(ABCD\) is a rhombus.
(i) From \(\triangle DAC\) and \(\triangle BCA\), show \(\angle DAC = \angle BCA\) and \(\angle ACD = \angle CAB\), etc.
(ii) Show \(\angle BAC = \angle BCA\), using Theorem 8.4.
\(ABCD\) is a rectangle in which diagonal \(AC\) bisects \(\angle A\) as well as \(\angle C\). Show that:
(i) \(ABCD\) is a square,
(ii) diagonal \(BD\) bisects \(\angle B\) as well as \(\angle D\).
In parallelogram \(ABCD\), two points \(P\) and \(Q\) are taken on diagonal \(BD\) such that \(DP = BQ\). Show that:
(i) \(\triangle APD \cong \triangle CQB\)
(ii) \(AP = CQ\)
(iii) \(\triangle AQB \cong \triangle CPD\)
(iv) \(AQ = CP\)
(v) \(APCQ\) is a parallelogram
\(ABCD\) is a parallelogram and \(AP\) and \(CQ\) are perpendiculars from vertices \(A\) and \(C\) on diagonal \(BD\). Show that:
(i) \(\triangle APB \cong \triangle CQD\)
(ii) \(AP = CQ\)
\(ABCD\) is a trapezium in which \(AB \parallel CD\) and \(AD = BC\). Show that:
(i) \(\angle A = \angle B\)
(ii) \(\angle C = \angle D\)
(iii) \(\triangle ABC \cong \triangle BAD\)
(iv) diagonal \(AC = BD\)