Difference between revisions of "1957 AHSME Problems/Problem 21"

Latest revision as of 19:55, 10 June 2024

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
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

@@ Line 18: / Line 18: @@
 (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. <math>\boxed{\textbf{(E) } (3), (4)}</math>
+(4) is a restatement of the original theorem.
+Therefore, (3) and (4) are correct.
+<math>\boxed{\textbf{(E) } (3), (4)}</math>
+==See Also==
+{{AHSME 50p box|year=1957|num-b=20|num-a=22}}
+{{MAA Notice}}
+[[Category:AHSME]][[Category:AHSME Problems]]