# Whats the difference between converse of alternate interior angle theorem and alternate interior angle theorem?

Jan 28, 2016

Consider two statements:
(A) Two lines that are cut by a transversal are parallel
(B) Alternate interior angle formed by these lines are congruent
They are equivalent.
See below for explanation.

#### Explanation:

Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent.

Let's represent it in a form "if A then B":
If two lines that are cut by a transversal are parallel [Part A] then alternate interior angles formed by these lines are congruent [Part B].

Converse theorem should look like "if B then A":
If alternate interior angles formed by these lines are congruent [Part B] then two lines that are cut by a transversal are parallel [Part A].

So, these are two different theorems, each requiring its own proof. But, since both theorem $A \to B$ and $B \to A$ can be proven independently, both statement are equivalent. If one is true, another is as well, if one if false, another is well.