QUESTION

# Which of the following always holds?(a) Every rectangle is a square(b) Every parallelogram is a trapezium(c) Every rhombus is a square(d) Every parallelogram is a rectangle

Hint: We need to understand the definitions of rectangle, square, parallelogram, trapezium and rhombus and check the given statements one by one to find the answer.

First we will understand the definitions of the following geometrical figures.

Quadrilateral- A closed figure having four sides and four vertices.

Rectangle- A quadrilateral in which all the angles between the sides are ${{90}^{\circ }}$.

Square- A square is a rectangle in which all the sides are of equal length.

Trapezium- A quadrilateral in which a pair of opposite sides are parallel.

Parallelogram- A quadrilateral in which opposite sides are parallel and the opposite angles are equal.

Rhombus- A parallelogram in which all the sides are of equal length.

Thus we may now check the statements one by one using the above definitions

(a) The given statement is that every rectangle is a square. This is not always true, because in a rectangle the sides can be of unequal length but all the sides in a square should be of equal length.

(b) The given statement is every parallelogram is a trapezium. This is a true statement, because trapezium is defined as a quadrilateral in which a pair of opposite sides are parallel. There is no constraint on the other pair of sides. As, in a parallelogram both pairs of opposite sides are equal, it will always satisfy the definition of a trapezium.

(c) The given statement is every rhombus is a square. It is not always true because in a rhombus the angles can be different from ${{90}^{\circ }}$ but in a square all the angles have to be equal to ${{90}^{\circ }}$.

(d) The given statement is that every parallelogram is a rectangle. It is not always true because in a rectangle, all the angles have to be ${{90}^{\circ }}$but in a parallelogram, the angles can be different from ${{90}^{\circ }}$.

Note: In the definitions, one should be careful that the constraints are put only on some of the properties of the quadrilateral. However, the other properties are free to assume any value and thus we can obtain different figures from the original one. For example, a square is also a rectangle having all sides equal.