Proofs using parallelism

Parallelism, combined with the congruence criteria, is an extremely powerful tool. Let us look at a classic proof tackled with backward reasoning (Backward Thinking): we start from the thesis and work back towards the hypotheses.

Example — Backward Thinking

$M$ is the midpoint of $PQ$. A line $r$ other than $PQ$ passes through $M$. On $r$ we take $S$ and $T$ on opposite sides of $M$ with $MS\cong MT$. Prove that $PT \parallel QS$.

The configuration: $M$ midpoint of $PQ$, and $S,T$ symmetric with respect to $M$ on a line through $M$.

The Backward Thinking method is visualised with a chain diagram that starts from the thesis and works back towards the hypotheses:

The chain diagram of backward reasoning: from the thesis (at the top) we descend down to the hypotheses (at the bottom).

Read from the bottom upwards, the diagram becomes the proof:

Proof

1. Hypothesis. $PM \cong QM$ (midpoint), $SM \cong TM$ (hypothesis), $\widehat{PMS}\cong\widehat{QMT}$ (vertically opposite).
2. Deduce. By the first criterion: $\triangle PMT\cong\triangle QMS$.
3. Deduce. Hence $\widehat{TPM}\cong\widehat{SQM}$ (corresponding elements).
4. Deduce. By the alternate interior angles theorem: $PT \parallel QS$.

$\blacksquare$

Links

Topics: Euclidean geometry
Concepts: Alternate interior angles · Congruence criteria · Proof · Parallelism · Line
Skills: Proving · Synthetic geometry