One common mistake in mathematics is assuming that if a statement is true, the converse of the statement is also true. To form the converse of a statement, you switch the "if" and "then" parts of the statement. Here's an example where the converse is clearly not true:
Statement: If you live in San Francisco, then you live in California. Converse: If you live in California, then you live in San Francisco.
You can write the Pythagorean theorem as an "if-then" statement as well:
The Pythagorean theorem: If, given a triangle, the square built on the hypotenuse is equal to the sum of the sqares built on the other two sides.
Converse: If the square built on the hypotenuse is equal to the sum of the squares built on the other two sides, then you have a right triangle. (If a2 + b2 = c2 for some triangle, then it must be a right triangle.)