Thales contrapositive: prove two lines are NOT parallel

Recap: direct Thales

With $(BC) \parallel (B'C')$ and $A, B, B'$ collinear, $A, C, C'$ collinear:

$\dfrac{AB}{AB'} = \dfrac{AC}{AC'} = \dfrac{BC}{B'C'}$

The contrapositive

If ratios are NOT equal, lines are NOT parallel.

$\text{If } \dfrac{AB}{AB'} \neq \dfrac{AC}{AC'}, \text{ then } (BC) \not\parallel (B'C')$

Don’t confuse with converse

Converse: ratios equal + right order ⇒ parallel. Contrapositive: “if NOT conclusion, then NOT hypothesis”.

Method to prove NOT parallel

1. Compute $\dfrac{AB}{AB'}$ and $\dfrac{AC}{AC'}$.
2. If different, conclude not parallel.
3. If equal, contrapositive doesn’t help. Use converse.

Example

$AB = 3$, $AB' = 5$, $AC = 6$, $AC' = 8$.

• $3/5 = 0.6$
• $6/8 = 0.75$

$0.6 \neq 0.75$ ⇒ contrapositive says: NOT parallel.

Common mistakes

1. Confusing contrapositive and converse.
2. Misaligned lengths: need $A, B, B'$ collinear, and $A, C, C'$ collinear.
3. Comparing without common denominator: $3/5$ vs $6/10$ → same!

?À toi de jouer

AB/AB' = 2/5, AC/AC' = 4/9. Can you conclude?

A.Yes, (BC) and (B'C') are NOT parallelB.Yes, they ARE parallelC.No conclusionD.Missing info

Valider