# Counterexamples

- Author:
- Simona Riva

- Topic:
- Congruence, Geometry, Rectangle, Square, Triangles

## What is a counterexample?

A

*counterexample*is an example in which the hypothesis is true, but the conclusion is false. If you can find a*counterexample*to a conditional statement, then that conditional statement is false. Using counterexamples to prove that a property or a theorem don't hold is sometimes easier that writing a complete proof. We will explore a couple of interactive examples below.## Congruence of triangles and union

Even if you never heard about the concept of congruence, you can easily recognize two congruent geometric objects, because they are identical in size and shape.
Use the applet below to create a is (and we write this as ) and a triangle is ( ), then the union of and ( ) is congruent to the union of and ( ).
You can

*counterexample*for the following statement: If a triangle*congruent*to a triangle*congruent*to a triangle*drag*the triangles by dragging the whole shape, or using the*red points*, and*rotate*them using the*green points*.## Rectangles with given area

Explore the applet below that shows (how many?) counterexamples for the following statement:

*If a rectangle has area*16,*then its perimeter is*20. Can you also show that there is a special case in which the rectangle is actually a square?