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 \(AC = BD\)
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. AC is a diagonal. Show that:
(i) SR ∥ AC and SR = 1/2 AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
Show PQRS is a parallelogram. Also show PQ ∥ AC and PS ∥ BD. So, ∠P = 90°.
ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
ABCD is a trapezium in which AB ∥ DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F. Show that F is the mid-point of BC.
In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.
AECF is a parallelogram. So, AF ∥ CE, etc.
ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that:
(i) D is the mid-point of AC
(ii) MD ⟂ AC
(iii) CM = MA = 1/2 AB