Question

Is parallelogram a rectangle? Can you call a rectangle a parallelogram?

Answer 1

Hint: From the given question, we have to discuss only the geometrical shapes of a rectangle and a parallelogram. First, we have to discuss some properties of a rectangle and properties of a parallelogram.

Complete step-by-step answer:
For the result, first we have to discuss some properties of a rectangle and properties of a parallelogram.
A rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal$\left({\displaystyle \frac{{360}^{\circ }}{4}}={90}^{\circ }\right)$. It can also be defined as a parallelogram containing a right angle.

Properties of a rectangle are given below:
$1.$ A rectangle has four sides, four vertices and four angles.
$2.$ Opposite sides are congruent. $$\overline{\text{AB}}\cong \overline{\text{DC}}$$, $$\overline{\text{AD}}\cong \overline{\text{BC}}$$.
$3.$ Adjacent sides are perpendicular.$$\overline{\text{AB}}\mathrm{\perp }\overline{\text{BC}}$$,$$\overline{\text{BC}}\mathrm{\perp }\overline{\text{CD}}$$,$$\overline{\text{CD}}\mathrm{\perp }\overline{\text{DA}}$$,$$\overline{\text{DA}}\mathrm{\perp }\overline{\text{AB}}$$.
$4.$ Opposite sides are parallel.$$\overline{\text{AB}}\parallel \overline{\text{DC}}$$,$$\overline{\text{AD}}\parallel \overline{\text{BC}}$$.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.

Properties of a parallelogram are given below:
$1.$ The diagonals of a parallelogram bisect each other.
$2.$ Opposite angles of a parallelogram are congruent.
$3.$ Opposite sides of a parallelogram are congruent.
$4.$ Each diagonal bisects the parallelogram into two congruent triangles. That is, any two adjacent angles add up to ${180}^{\circ }$.
$\mathrm{\angle }\text{A}+\mathrm{\angle }\text{B}={180}^{\circ }$, $$\mathrm{\angle }\text{B}+\mathrm{\angle }\text{C}={180}^{\circ }$$,$$\mathrm{\angle }\text{C}+\mathrm{\angle }\text{D}={180}^{\circ }$$,$\mathrm{\angle }\text{D}+\mathrm{\angle }\text{A}={180}^{\circ }$.
From the given question, now we are going to discuss only the geometrical shapes of a rectangle and a parallelogram.
From the above discussed properties of a rectangle and properties of a parallelogram, we are going to conclude the required first result in the following manner.
The main characteristic of a rectangle is that the opposite sides must be equal and all the angles must be ${90}^{\circ }$. A parallelogram has opposite equal sides but the angles may not be ${90}^{\circ }$.
$\therefore$ Every parallelogram cannot be rectangle.
Now, we are going to conclude the second result in the following manner.
The main characteristic of a parallelogram is that the opposite sides must be parallel to each other. In a rectangle the opposite sides are parallel to one another.
$\therefore$ Every rectangle is a parallelogram.
Therefore, we have to conclude the required result.

Yes, every parallelogram cannot be a rectangle but every rectangle is a parallelogram.

Note: A square is similar to a rectangle. Square has 4 sides and 2 internal diagonals with internal angles equal to 90. The only difference between square and rectangle is that, in a square all the sides are equal but in a rectangle only the opposite sides are equal.