# 1959 AHSME Problems/Problem 6

## Problem 6

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 is: $\textbf{(A)}\ \text{only the converse is true} \qquad \\ \textbf{(B)}\ \text{only the inverse is true }\qquad \\ \textbf{(C)}\ \text{both are true} \qquad \\ \textbf{(D)}\ \text{neither is true} \qquad \\ \textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true}$

## Solution

First, let us list the statement "If a quadrilateral is a square, then it is a rectangle" as a statement of the form "If $p$, then $q$". In this case, $p$ is "a quadrilateral is a square", and $q$ is "it is a rectangle".

The converse is then: "If $q$, then $p$". Plugging in, we get "If a quadrilateral is a rectangle, then it is a square". However, this is obviously false, as a rectangle does not have to have four sides of the same measure.

The inverse is then: "If not $p$, then not $q$". Plugging in, we get "If a quadrilateral is not a square, then it is not a rectangle". This is also false, since a quadrilateral can be a rectangle but not be a square.

Therefore, the answer is $\boxed{\textbf{D}}$.

## See also

 1959 AHSC (Problems • Answer Key • Resources) Preceded byProblem 5 Followed byProblem 7 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 All AHSME Problems and Solutions

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