1999 AHSME Problems/Problem 2

Problem

Which of the following statements is false?

$\mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}$

$\mathrm{(B) \  All\ equilateral\ triangles\ are\ convex.}$

$\mathrm{(C) \  All\ equilateral\ triangles\ are\ equianguilar.}$

$\mathrm{(D) \  All\ equilateral\ triangles\ are\ regular\ polygons.}$

$\mathrm{(E) \  All\ equilateral\ triangles\ are\ similar\ to\ each\ other.}$

Solution

An equilateral triangle is isosceles, and we find that $\angle A=\angle B=\angle C$ if we use the property of isosceles triangles that if two sides of a triangle are equal then the opposite angles are equal. Thus equilateral triangles are equiangular, and $C$ is true.

Regular polygons are both equilateral and equiangular, and so are equilateral triangles are both equilateral (by definition) and equiangular (by the above argument). Thus equilateral triangles are regular polygons and $D$ is true.

Since all of the angles in an equilateral triangle are congruent, all equilateral triangles are similar by AAA similarity. Thus, $E$ is true.

Since $\angle A=\angle B=\angle C$ and $\angle A+\angle B+\angle C=180$, $\angle A=60^{\circ}$. Since no other angles are above $180^{\circ}$, all equilateral triangles are convex. Thus, $B$ is true.

This just leaves choice $\boxed{\mathrm{(A)}}$. This is clearly false: an equilateral triangle with side $1$ is not congruent to an equilateral triangle with side $2$.

See also

1999 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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