IFFY Triangles

This problem encourages students to think about the directions of implication ("if" and "only if").  They can also be introduced to the $\implies$, $\Longleftarrow$, and $\iff$ notation.  Question $1$ uses the idea of an arithmetic progression, and question $2$ can be solved by using congruent triangles.  The results of question $2$ should be familar to students, but they might not have met a proof of this result before.

Here are some statements that can be used to introduce "if" and "only if" (or "implies" and "is implied by") notation.  In each case students should discuss which are correct and which are not correct.  

• Tiddles is a cat $\Longrightarrow$ Tiddles is not a horse
  Tiddles is a cat $\Longleftarrow$ Tiddles is not a horse
   
• $n$ is odd $\Longrightarrow$ $n^2$ is odd
  $n$ is odd $\Longleftarrow$ $n^2$ is odd
   
• $x^2=1$ $\Longrightarrow$ $x=1$
  $x^2=1$ $\Longleftarrow$ $x=1$
   
• $n$ is a multiple of $4$ $\Longrightarrow$ $n^2$ is a multiple of $4$
  $n$ is a multiple of $4$ $\Longleftarrow$ $n^2$ is a multiple of $4$

For the second example both directions are true so we can write "$n$ is odd $\iff$ $n^2$ is odd".  Students can be challenged to come up with some examples of their own, and there are more examples in the problem Iffy Logic.

Possible Extension - students might like to consider Euclid's "Pons Asinorum" (bridge of donkeys) proof that a triangle with two equal sides has two equal angles.  In this proof, the congruent triangle condition $SSS$ cannot be assumed (as Euclid had not proved this at this point!).  This proof is explored in question 4 of this STEP Support Programme Assignment.