Worksheets › Class 9 › Propositions and their Converses › Cheat sheet
Class 9 Propositions and their Converses: cheat sheet
CBSE Mathematics. 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.
Remember
| 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 |
Watch out
- Assuming the converse is true because the statement is.
- Giving an example instead of a counterexample when asked to disprove.
- Comparing a² + b² with c² when c is not the longest side.
Worked example
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
Practise Propositions and their Converses
Revise with this page, then practise: every Propositions and their Converses sheet in the app is new, with an answer key and working.
Coming soon for Android and iPhone.