P: Two circles are congruent if and only if their radii are equal. Check the validity of statement P

Explanation

In order to prove the statement "p if and only if q", we need to show that
(i) If p is true, then q is true and (ii) If q is true, then p is true or show that `~q rArr ~p`.
Now, Let p : Two circles are congruent.
q : Radius of two circles are equal.
Assume that pis true i.e., circles are congruent, we know that two figures are congruent thentheir areas are also equal. Let radius of two circles be `r_1 and r_2`
Now, Area of circles are equal
So, `pir_1^2=pir_2^2`
`rArr r_1^2=r_2^2`
`rArr r_1=r_2`
So, their radii are equal , so q is also true
Now, soppose q is true, i.e., radius of both these circles are equal. When radius of both these circles are equal then they will of same size. So, two circles are of same shape and size which i is the required criteria of congruency. Hence, both circles are congruent. So p is also true.