# Implications

If two statements are connected by “if-then”, “only if” and “if and only if”, they are called implications.Such statements are very common in mathematics.

**Example 1:**

If a triangle is right angled, then the square on the hypotenuse is equal to the sum of the squares on the other two sides.

**Solution:**

: A triangle is right angled

: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

We have combined these statements by “if-then” implication. The resulting statement is

*⇒ ; which means if*

*, then*

*.*

**Example 2:**

A quadrilateral is a parallelogram only if the diagonals bisect each other.

**Solution:**

p: a quadrilateral is a parallelogram

*q*: diagonals bisect each other.

We have combined the two statements with “only if” implication.

i.e. the 1

^{st}part is dependant on the 2

^{nd}part or it is equivalent to

*q*⇒

*p*.

**Example 3:**

Two triangles are congruent if and only if their corresponding sides are equal.

**Solution:**

*p*: Two triangles are congruent

*q*: Their corresponding sides are equal

This is a two way implication

i.e.

*p*⇒

*q*is if then

and

*p*⇒

*q*is only if

*p*⇒

*q*is if and only if.