# Given below are some pairs of statements. Combine each pair using if and only if :

Question:

Given below are some pairs of statements. Combine each pair using if and only if :

(i) $p$ : If a quadrilateral is equiangular, then it is a rectangle.

$q$ : If a quadrilateral is a rectangle, then it is equiangular.

(ii) $p$ : If the sum of the digits of a number is divisible by 3 , then the number is divisible by 3 .

$q$ : If a number is divisible by 3 , then the sum of its digits is divisible by $3 .$

(iii) $\mathbf{p}: \mathbf{A}$ quadrilateral is a parallelogram if its diagonals bisect each other.

$q$ : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

(iv) $p$ : If $f(a)=0$, then $(x-a)$ is a factor of polynomial $f(x)$.

$q:$ If $(x-a)$ is a factor of polynomial $f(x)$, then $f(a)=0$.

(v) $p$ : If a square matrix $A$ is invertible, then $|A|$ is nonzero.

$q$ : If $A$ is a square matrix such that $|A|$ is nonzero, then $A$ is invertible.

Solution:

(i) A quadrilateral is a rectangle if and only if it is equiangular.

(ii) A number is divisible by 3 if and only if the sum of the digits of the number is divisible by 3 .

(iii) A quadrilateral is a parallelogram if and only if its diagonals bisect each other.

(iv) $(x-a)$ is a factor of polynomial $f(x)$ if and only if $f(a)=0$.

(v) Square matrix $A$ is invertible if and only if $|A|$ is nonzero.