Difference between revisions of "1957 AHSME Problems/Problem 21"
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1. If two angles of a triangle are not equal, the triangle is not isosceles. | 1. If two angles of a triangle are not equal, the triangle is not isosceles. | ||
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2. The base angles of an isosceles triangle are equal. | 2. The base angles of an isosceles triangle are equal. | ||
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3. If a triangle is not isosceles, then two of its angles are not equal. | 3. If a triangle is not isosceles, then two of its angles are not equal. | ||
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4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles. | 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? | Which combination of statements contains only those which are logically equivalent to the given theorem? | ||
| − | <math>\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 </math> | + | <math>\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 </math> |
==Solution== | ==Solution== | ||
Revision as of 15:35, 10 June 2024
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?
Solution
(1) is the inverse
(2) is the converse
(3) is the contrapositive
(4) is a restatement of the original conditional
Therefore, (3) and (4) are correct. 
