Propositions and converses

A proposition is a statement that is either true or false, like 'if two sides of a triangle are equal, the angles opposite them are equal'.

Many propositions have the shape 'if X, then Y'. The same thing can be said as 'X implies Y' or 'Y when X'.

Swap the two parts and you get the converse: 'if Y, then X'. 'If two angles of a triangle are equal, the sides opposite them are equal' is the converse of the statement above.

A true statement can have a false converse.

• 'If it rains, then the road is wet': true.
• 'If the road is wet, then it has rained': not always. A water tanker may have spilled.

With numbers: 'if a number is a multiple of 6, it is a multiple of 3' is true, because 6k = 3 × 2k. The converse is false: 9 is a multiple of 3 but not of 6.

Sometimes the converse is true as well. Equal angles do give equal sides. Draw the altitude from the third corner: the two halves are congruent (AAS). But it had to be proved separately: the first statement being true did not make it true.