What Is a Counterexample in Geometry?

A counterexample, in geometry as in other areas of mathematics and logic, is an example that one uses to prove that a particular statement is false. A simple example from primary mathematics uses the statement “the inverse of a number is never an integer,” and its counterexample would be 1/4. The inverse of 1/4 is 4, which is an integer. For geometry, finding counterexamples involves a few more calculations.

Geometry is primarily concerned with the formation and interaction of shapes, as well as with the development of logical skills to write proofs of various mathematical facts. An individual can consider the “Similar triangles have at least one side that is congruent.” While triangles that are similar do have congruent angles, none of their sides are congruent. Instead, they have to remain in proportion with one another to keep the angles congruent.

One can consider triangle ABC with sides of 8, 8 and 4. The fact that two of the sides are congruent to each other means that two angles are as well, making this an isosceles triangle. Triangle DEF has sides of 8, 16 and 2. Using a protractor reveals that, even though one side of DEF is congruent to one side of ABC, the angles are not close to being the same, so the triangle is not similar. However, in triangle XYZ, that has sides of 4, 4 and 2, the proportional relationships between the three sides of ABC and XYZ are all 2-to-1. Because the proportions are the same, the angles are as well, making the two triangles similar.