1957 AHSME Problems/Problem 21

Problem 21

Start with the theorem "If two angles of a triangle are equal, the triangle is isosceles," and the following four statements:

1. If two angles of a triangle are not equal, the triangle is not isosceles.

2. The base angles of an isosceles triangle are equal.

3. If a triangle is not isosceles, then two of its angles are not equal.

4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles.

Which combination of statements contains only those which are logically equivalent to the given theorem?

$\textbf{(A)}\ 1,\,2,\,3,\,4 \qquad \textbf{(B)}\ 1,\,2,\,3\qquad \textbf{(C)}\ 2,\,3,\,4\qquad \textbf{(D)}\ 1,\,2\qquad\textbf{(E)}\ 3,\,4$

(1) is the inverse

(2) is the converse

(3) is the contrapositive

(4) is a restatement of the original theorem.

Therefore, (3) and (4) are correct. $\boxed{\textbf{(E) } (3), (4)}$

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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