A [i]counterexample[/i] is an example in which the hypothesis is true, but the conclusion is false. [br]If you can find a [i]counterexample [/i]to a conditional statement, then that conditional statement is false.[br][br]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.

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.[br][br]Use the applet below to create a [i]counterexample [/i]for the following statement:[br]If a triangle [math]\large{T}[/math] is [i]congruent [/i]to a triangle [math]\large{T'}[/math](and we write this as [math]\large{T\cong T'}[/math]) and a triangle [math]\large{S} [/math] is [i]congruent [/i]to a triangle [math]\large{S'}[/math] ([math]\large{S\cong S'}[/math]), then the union of [math]\large{T}[/math] and [math]\large{S}[/math] ([math]\large{T\cup S}[/math]) is congruent to the union of [math]\large{T'}[/math] and [math]\large{S'}[/math] ([math]\large{T'\cup S'}[/math]).[br][br]You can [i]drag [/i]the triangles by dragging the whole shape, or using the [i][color=#ff0000]red points[/color][/i], and [i]rotate [/i]them using the [i][color=#38761d]green points[/color][/i].

Explore the applet below that shows (how many?) counterexamples for the following statement:[br][br][i]If a rectangle has area[/i] 16, [i]then its perimeter is[/i] 20.[br][br]Can you also show that there is a special case in which the rectangle is actually a square?