1957 AMC 12 Problem 21

Question

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?

1,

2,

3,

4

1,

2,

3

2,

3,

4

1,

2

3,

4

Write the theorem as

P\Rightarrow Q

and compare each statement with its inverse, converse, or contrapositive

A statement is always equivalent to its contrapositive, but not generally to its converse or inverse

Let

P

mean that two angles are equal and

Q

that the triangle is isosceles. Statement

3

is the contrapositive

\neg Q\Rightarrow\neg P,

so it is equivalent to

P\Rightarrow Q.

Statement

4

says that

Q

is necessary for

P,

which is another wording of

P\Rightarrow Q.

Statements

1

and

2

are the inverse and converse.

Thus only statements

3

and

4

are equivalent, so the correct answer is E.

Problem 21 in other years