(i) False

Reason: Imagine a parallelogram and draw its diagonals. Now the areas of the two triangles on one of the bases is equal. But by Heron's formula, the areas are not equal and if the areas are not equal how can be the diagonals because area can only be equal if both the triangles have equal diagonals.

(ii) True

Proof:

Let’s take a parallelogram ABCD,

The diagonals AC and BD intersect each other at O,

AO = OC and BO = OD

In ΔAOB and ΔCOD,

We have

∠BAO = ∠OCD (alternate interior angles)

∠AOB = ∠CDO (alternate interior angles)

AB = DC (opposite sides)

ΔAOB≅ΔCOD (by ASA)

AO = OC and DO = OB

(iii) False

According to the definition a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of aparallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

It turns out that a parallelogram has its diagonalsmeeting at right angles if and only if the parallelogram is a rhombus (all sides equal).

(iv) True

Reason: Let’s take ABCD is a quadrilateral,

Where AB = CD & AD = BC

We have to prove : ABCD is a parallelogram

Join AC, it’s a diagonal

In ΔABC and ΔCDA

AB = CD

BC = DA

AC = CA

ΔABC≅ΔCDA

Hence,

∠BAC = ∠DCA

For lines AB and CD with transversal AC,

∠BAC & ∠DCA are alternate angles and are equal.

So, AB and CD lines are parallel.

∠BCA = ∠DAC

For lines AD and BC with transversal AC,

∠BCA & ∠DAC are alternate angles and are equal.

So, AD and BC lines are parallel.

Thus in ABCD,

Both pairs of opposite sides are parallel,

So, ABCD is a parallelgram.

(v) False

Reason: If all the angles of a quadrilateral are equal, then it’s a rectangle.

(vi) False

Reason:

As we know that the opposite or facing sides of aparallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

In case of square and rhombus all side are equal, but if three sides are equal then it doesn’t satisfied the property of parallelogram.

(vii) False

Reason:

Same reason as for the sides, it does not satisfied the property of a parallelogram.

(viii) True

Yes if all side of a parallelogram than it’s a square or rhombus.