1959 AMC 12 Problem 6

Question

Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement:

only the converse is true

only the inverse is true

both are true

neither is true

the inverse is true, but the converse is sometimes true

Answer: D

Concepts:logical deduction counterexample

Difficulty rating: 1140

Small hint:

Write the converse and inverse explicitly

Big hint:

Use a nonsquare rectangle to test the converse and a nonrectangular nonsquare to test the inverse

Solution:

The converse says that every rectangle is a square, which is false. The inverse says that every quadrilateral that is not a square is not a rectangle, which is also false because a nonsquare rectangle is a counterexample.

Thus neither statement is true, and the correct answer is D.