Propositions and their Converses

A proposition is true or false. Swapping its hypothesis and conclusion gives the converse, which may or may not be true.

Key ideas

A proposition is a statement that is either true or false, never both.
In “If P, then Q”, P is the hypothesis and Q the conclusion.
The converse of “If P, then Q” is “If Q, then P”. A true statement can have a false converse.
One counterexample is enough to prove a general statement false; many examples never prove it true.
When a statement and its converse are both true we say “P if and only if Q”.
Converse theorems are tests: equal alternate angles show lines are parallel; a² + b² = c² shows a right angle.

Converse of Pythagoras	a² + b² = c² ⇒ right angle opposite c; > acute; < obtuse
Isosceles converse	two equal angles ⇒ the sides opposite them are equal
Mid-point converse	line through the midpoint of one side, parallel to another, bisects the third
Parallel lines	co-interior angles add to 180° ⇒ the lines are parallel

Show by a counterexample that 'If a number is divisible by 6 and by 9, then it is divisible by 54' is false.

Working: 180 = 6 × 30 = 9 × 20, but 180 ÷ 54 leaves remainder 18; it works because 6 and 9 share the factor 3

Answer: 180 is divisible by 6 and 9, but not by 54