Propositions and their Converses Class 9: Converses, Counterexamples and Proofs

What Propositions and their Converses is about

Weightage: This chapter is part of Unit IV, Geometry, which carries 25 of the 80 marks in the school annual exam, shared with Introduction to Euclid's Geometry: Axioms and Postulates, Lines and Angles, 4-gons (Quadrilaterals) and Circles.Source: CBSE Mathematics syllabus, Class IX 2026-27, course structure, Unit IV Geometry

Key concepts

What a proposition is

NCERT section Propositions and their Converses

A proposition is a statement that is definitely true or false, even if we do not yet know which.

• 'Two sides of a triangle are equal, so the opposite angles are equal' is a proposition (true).
• '2 + 2 = 5' is a proposition (false).
• 'Close the window' is an instruction, and 'What time is it?' is a question; neither is true or false, so neither is a proposition.
• 'x + 3 = 5' is not a proposition until we know what x is.

Worked example

Which of these are propositions? For each proposition, say whether it is true or false. (i) 7 is a prime number. (ii) Please sit down. (iii) Every square is a rectangle. (iv) 15 is a multiple of 4. (v) Is it raining?

Try it first, then show the solutionSolution

1. (i) A statement that is true: a proposition, true.
2. (ii) An instruction: not a proposition.
3. (iii) A statement; every square has four right angles: a proposition, true.
4. (iv) A statement; $15÷4$ is not whole: a proposition, false.
5. (v) A question: not a proposition.

Answer: (i) true, (iii) true and (iv) false are propositions; (ii) and (v) are not.

If-then statements and other ways of saying them

NCERT section Propositions and their Converses

Many propositions have the form 'if X then Y': X is the condition (hypothesis) and Y is what follows (conclusion).

The same meaning can be written in other ways:

• 'X implies Y'
• 'Y when X' or 'Y whenever X'
• 'Every X is Y' (for example, 'every multiple of 10 is a multiple of 5' says the same as 'if a number is a multiple of 10, then 5 divides it')

Before writing a converse, rewrite the statement in clear if-then form so you can see which part is X and which is Y.

Worked example

Rewrite in 'if X then Y' form, and name X and Y: (i) Every multiple of 10 is a multiple of 5. (ii) A quadrilateral has four right angles when it is a rectangle.

Try it first, then show the solutionSolution

1. (i) If a number is a multiple of 10, then 5 divides it. X: multiple of 10; Y: multiple of 5.
2. (ii) 'Y when X' means 'if X then Y': If a quadrilateral is a rectangle, then all four of its angles are right angles. X: rectangle; Y: four right angles.

Answer: (i) If a multiple of 10, then a multiple of 5. (ii) If a rectangle, then four right angles.

The converse of a proposition

NCERT section Propositions and their Converses

The converse of 'if X then Y' is 'if Y then X': swap the condition and the conclusion.

When a statement begins with a fixed setting (for example, 'Let a, b, c be the sides of a triangle.'), keep that setting and swap only the if-part and the then-part.

A proposition being true tells you nothing about whether its converse is true. You must check the converse separately.

Worked example

Write the converse of each: (i) If two lines are parallel, then the corresponding angles made by a transversal are equal. (ii) If a quadrilateral is a square, then all its angles are equal.

Try it first, then show the solutionSolution

1. (i) Converse: If the corresponding angles made by a transversal with two lines are equal, then the lines are parallel.
2. (ii) Converse: If all the angles of a quadrilateral are equal, then it is a square.

Answer: (i) Equal corresponding angles imply parallel lines (true). (ii) Equal angles imply a square (false: a rectangle that is not a square has all angles equal).

Counterexamples

NCERT section Propositions and their Converses

A counterexample is one case where the condition holds but the conclusion fails. One counterexample is enough to show that a general statement is false; no number of examples where it works can prove it true.

Famous case: Fermat claimed every number of the form $2^{2^n}+1$ is prime. It works for $n=0,1,2,3,4$ (3, 5, 17, 257, 65537), but Euler showed $2^{2^5}+1=4294967297=641\times 6700417$.

Everyday case: 'If the road is wet, then it has rained' fails when a water tanker has spilled water on a dry day.

Worked example

Find a counterexample to each claim: (i) $n^2-n+11$ is prime for every natural number n. (ii) If n is a prime number, then $2n+1$ is prime.

Try it first, then show the solutionSolution

1. (i) Try values: $n=1$ gives 11, $n=2$ gives 13, … but $n=11$ gives $121-11+11=121=11\times 11$, not prime.
2. (ii) $n=7$ is prime, but $2(7)+1=15=3\times 5$ is not prime.

Answer: (i) $n=11$ gives 121 (ii) $n=7$ gives 15

Four possibilities for a proposition and its converse

NCERT section Propositions and their Converses

Proposition	Converse	Example
true	false	If a number is a multiple of 6, it is a multiple of 3. (Converse fails: 9.)
true	true	If a triangle is equilateral, all its angles are 60°. (Converse also true.)
false	true	If two triangles have equal areas, they are congruent. (Converse 'congruent ⇒ equal areas' is true.)
false	false	If a number is odd, it is prime. (Fails at 9; the converse fails at 2.)

When both a statement and its converse are true, we say X and Y imply each other (X if and only if Y).

Worked example

Proposition: If two triangles are congruent, then they have the same area. Write the converse and decide whether each is true.

Try it first, then show the solutionSolution

1. Proposition: congruent triangles fit exactly on each other, so their areas are equal. True.
2. Converse: If two triangles have the same area, then they are congruent.
3. Counterexample: a triangle with base 6 and height 2 and one with base 4 and height 3 both have area 6, but their sides differ, so they are not congruent. The converse is false.

Answer: Proposition true; converse false (counterexample: base 6 height 2, and base 4 height 3).

Proving a converse: the isosceles triangle theorem

NCERT section Propositions and their Converses

Proposition: if two sides of a triangle are equal, then the angles opposite them are equal.
Converse: if two angles of a triangle are equal, then the sides opposite them are equal. It is also true; the proof uses an altitude.

In $△ABC$ with $\angle B=\angle C$, draw the altitude AD to BC. In triangles ABD and ACD: $\angle B=\angle C$ (given), $\angle ADB=\angle ADC=90°$, and AD is common. So $△ABD\cong △ACD$ (AAS), and $AB=AC$ (corresponding parts).

Worked example

In $△PQR$, $\angle Q=\angle R=65°$ and $PQ=7$ cm. Find PR and $\angle P$, naming the result you use.

Try it first, then show the solutionSolution

1. By the converse of the isosceles triangle theorem, equal angles at Q and R mean $PR=PQ=7$ cm.
2. $\angle P=180°-65°-65°=50°$.

Answer: $PR=7$ cm and $\angle P=50°$.

The converse of the Baudhāyana–Pythagoras theorem

NCERT section Propositions and their Converses

Theorem: let a, b, c be the sides of a triangle; if the triangle is right-angled (with c opposite the right angle), then $a^2+b^2=c^2$.
Converse: if $a^2+b^2=c^2$, then the triangle is right-angled (at the angle opposite c).

Proof idea: build a second triangle XYZ with a right angle at Z and legs a and b. By the theorem, its hypotenuse is $\sqrt{a^2+b^2}=c$. Now both triangles have sides a, b, c, so they are congruent (SSS), and the first triangle also has a right angle.

Use: to test whether three lengths make a right triangle, check whether the square of the longest equals the sum of the squares of the other two.

Worked example

Do the lengths (i) 9, 40, 41 and (ii) 8, 15, 18 make right-angled triangles?

Try it first, then show the solutionSolution

1. (i) $9^2+40^2=81+1600=1681=41^2$. By the converse, yes, with the right angle opposite 41.
2. (ii) $8^2+15^2=64+225=289\ne 324=18^2$, so no.

Answer: (i) yes (ii) no

When a statement and its converse are both true

NCERT section Propositions and their Converses

Sometimes X and Y imply each other. Then two separate arguments are needed: one for 'X implies Y' and one for 'Y implies X'. Proving one direction does not prove the other.

NCERT's example: for a positive integer n, being a perfect square and having an odd count of factors each imply the other. Factors come in partner pairs that multiply to n (for 12: 1 × 12, 2 × 6, 3 × 4). The count is odd only if some factor is its own partner, $f\times f=n$, which means n is a square; and a square has exactly one such repeated pair, so its count is odd.

Worked example

List the factors of 36 and of 40. Use them to check the statement 'n has an odd number of factors exactly when n is a perfect square'.

Try it first, then show the solutionSolution

1. 36: pairs (1, 36), (2, 18), (3, 12), (4, 9), (6, 6). Factors 1, 2, 3, 4, 6, 9, 12, 18, 36: nine, odd; 6 pairs with itself.
2. 40: pairs (1, 40), (2, 20), (4, 10), (5, 8). Eight factors, even; no factor pairs with itself.
3. 36 is a square with an odd count; 40 is not a square and has an even count. Both fit the statement.

Answer: 36 has 9 factors (square); 40 has 8 factors (not a square).

Maths in real life: Propositions and their Converses

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• If-Then Statements, Converses and Counterexamples
• Isosceles Triangles and the Converse of the Pythagoras Theorem

Propositions and their Converses formula sheet

Proposition

$\text{a statement that is either true or false}$

Implication

$X\Rightarrow Y(\text{if }X\text{ then }Y)$

Also read 'X implies Y' or 'Y when X'.

Converse

$Y\Rightarrow X$

Swap the condition and the conclusion.

Counterexample to X ⇒ Y

$\text{a case with }X\text{ true and }Y\text{ false}$

One counterexample disproves a general statement.

Both directions

$X\iff Y(X\Rightarrow Y\text{ and }Y\Rightarrow X)$

Needs two separate justifications.

Isosceles theorem and converse

$AB=AC\iff \angle B=\angle C$

Converse proved with an altitude and AAS.

Baudhāyana–Pythagoras and converse

$\angle C=90^{\circ }\iff a^2+b^2=c^2$

c is the side opposite C.

Perfect squares

$n\text{ is a perfect square}\iff n\text{ has an odd number of factors}$

How the school exam asks Propositions and their Converses

The question patterns to expect, and the method that works for each.

Tricks to remember Propositions and their Converses

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Flip the arrow

  Converse = turn the arrow round: X ⇒ Y becomes Y ⇒ X.

  Use it when: Writing converses quickly.

• Rain makes roads wet, not the other way

  If it rains, the road is wet (true); if the road is wet, it rained (not always: a tanker!).

  Use it when: Remembering that a true statement can have a false converse.

• One bad apple

  One counterexample spoils the whole 'for all' claim.

  Use it when: Disproving general statements.

• Every square is a rectangle, not every rectangle a square

  Bigger family on the right, smaller family on the left: 'if square then rectangle' is true, but the converse fails.

  Use it when: Quadrilateral converse questions.

Propositions and their Converses: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Propositions and their Converses

The slips examiners see most often, and what to write instead.

• ✗Mistake: Swapping only part of the statement

  The setting and the condition get mixed up.

  Wrong working

  Converse of 'Let a, b, c be the sides of a triangle; if it is right-angled then $a^2+b^2=c^2$': 'If a, b, c are not sides of a triangle, then …'

  Right working

  Keep 'Let a, b, c be the sides of a triangle' and swap the rest: 'if $a^2+b^2=c^2$, then it is right-angled.'

  ✓Do this: Underline X and Y first; keep everything else.

  Usually costs: The converse mark.

• ✗Mistake: Proving a statement with examples

  Several cases that work feel like a proof.

  Wrong working

  12, 18 and 24 are multiples of 6 and of 3, so every multiple of 6 is a multiple of 3.

  Right working

  A multiple of 6 is $6k=3(2k)$, which is a multiple of 3 for every k.

  ✓Do this: Give a reason that covers every case.

  Usually costs: The justification mark.

• ✗Mistake: A counterexample that does not meet the condition

  Students forget that X must be true in the counterexample.

  Wrong working

  To disprove 'if n is a multiple of 3, it is a multiple of 6': take $n=5$.

  Right working

  Take $n=9$: it is a multiple of 3 but not of 6.

  ✓Do this: Check that your example satisfies the 'if' part and fails the 'then' part.

  Usually costs: The counterexample mark.

• ✗Mistake: Assuming the converse is true because the proposition is

  The two statements sound alike.

  Wrong working

  If a quadrilateral is a square, its angles are equal; so equal angles mean a square.

  Right working

  A 6 cm by 4 cm rectangle has equal angles but is not a square; the converse is false.

  ✓Do this: Always test the converse on its own.

  Usually costs: 1 to 2 marks.

NCERT exercise guide: Propositions and their Converses

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Propositions and their Converses

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Which of the following is NOT a proposition?
   (a) 0 is an even number.
   (b) A triangle has four sides.
   (c) Draw a circle of radius 3 cm.
   (d) $\sqrt{2}$ is irrational.

   Hint

   Can it be true or false?

   Answer

   (c), an instruction

2. Q2 · 1 mark · MCQExam level

   A counterexample to 'every prime number is odd' is
   (a) 9
   (b) 2
   (c) 15
   (d) 1

   Hint

   It must be prime and not odd.

   Answer

   (b) 2

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): 'If a quadrilateral is a square, then all its sides are equal' is true, but its converse is false.
   Reason (R): A rhombus that is not a square has all its sides equal.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

   Hint

   Is a non-square rhombus a counterexample?

   Answer

   (a)

4. Q4 · 2 marks · Very short answerExam level

   Write the converse of 'If a triangle is equilateral, then each of its angles is 60°.' Is the converse true?

   Hint

   Equal angles give equal sides.

   Answer

   If each angle of a triangle is 60°, it is equilateral. True: all angles equal means all opposite sides equal.

5. Q5 · 2 marks · Very short answerHarder exam level

   Find counterexamples: (i) $4n+3$ is prime for every natural number n. (ii) If n is even, then $2n+1$ is prime.

   Hint

   Try small values.

   Answer

   (i) $n=3$: 15 (ii) $n=4$: 9

6. Q6 · 3 marks · Short answerHarder exam level

   Proposition: If a and b are perfect squares, then ab is a perfect square. Justify it, write the converse and test it.

   Hint

   Try a = 2, b = 8.

   Answer

   If $a=m^2$, $b=n^2$, then $ab=(mn{)}^2$: true. Converse: if ab is a perfect square, then a and b are perfect squares. False: $2\times 8=16$ but 2 and 8 are not squares.

7. Q7 · 3 marks · Short answerHarder exam level

   Proposition: If $x=y$, then $a+x=a+y$ (for any numbers a, x, y). Write the converse and say whether it is true. Why are both used when solving equations?

   Hint

   Subtract a.

   Answer

   Converse: if $a+x=a+y$, then $x=y$; true (subtract a from both sides). Adding or subtracting the same number keeps an equation's solutions the same in both directions, so no solutions are lost or gained.

8. Q8 · 3 marks · Short answerHarder exam level

   Proposition: If n and n + 3 have no common factor other than 1, then n is not a multiple of 3. Is it true? Write the converse and test it.

   Hint

   If n were a multiple of 3, what would n + 3 be?

   Answer

   True: if n were a multiple of 3, so would $n+3$ be, giving the common factor 3. Converse: if n is not a multiple of 3, then n and n + 3 share no factor except 1. Also true: a common factor must divide their difference 3, so it could only be 3, which does not divide n.

9. Q9 · 5 marks · Long answerHarder exam level

   Prove that if $a^2+b^2=c^2$ for the sides of a triangle, the triangle is right-angled. Then check whether 7, 24, 25 and 6, 9, 11 form right triangles.

   Hint

   Build a right triangle with legs a and b, use SSS.

   Answer

   Build $△XYZ$ with right angle Z and legs a, b; its hypotenuse is $\sqrt{a^2+b^2}=c$; the triangles are congruent (SSS), so the original has a right angle. $49+576=625=25^2$: right. $36+81=117\ne 121$: not right.

10. Q10 · 2 marks · Very short answerHarder exam level

    Express the divisibility test for 3 as two if-then statements, and say whether both are true.

    Hint

    Number divisible by 3, digit sum divisible by 3.

    Answer

    If a number is divisible by 3, then the sum of its digits is a multiple of 3; if the sum of its digits is a multiple of 3, then the number is divisible by 3. Both are true, so the two facts imply each other.

11. Q11 · 3 marks · Short answerHOTS / competency

    Give an example of a proposition and its converse that are both false, with counterexamples.

    Hint

    Think of odd numbers and primes.

    Answer

    'If a number is odd, then it is prime' (false: 9). Converse: 'If a number is prime, then it is odd' (false: 2).

12. Q12 · 3 marks · Short answerHOTS / competency

    An advert says: 'Every champion drinks GoFast juice.' Rewrite it as an if-then statement and write its converse. Even if the advert is true, does drinking GoFast make you a champion? Explain.

    Hint

    Which way does the arrow point?

    Answer

    If a person is a champion, then they drink GoFast. Converse: if a person drinks GoFast, then they are a champion. The advert claims only the first; the converse does not follow, and millions who drink the juice are not champions.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What is a proposition?Show answer

   A statement that is either true or false.

2. 2.What is the converse of 'if X then Y'?Show answer

   'If Y then X'.

3. 3.What is a counterexample?Show answer

   A case where the condition is true but the conclusion is false, showing a statement is false.

4. 4.Can a true proposition have a false converse?Show answer

   Yes, for example 'if a multiple of 6, then a multiple of 3' (converse fails at 9).

5. 5.Who disproved Fermat's claim about numbers $2^{2^n}+1$?Show answer

   Euler, who showed $2^{32}+1$ is divisible by 641.

6. 6.Which congruence rule proves the converse of the isosceles theorem?Show answer

   AAS, using the altitude from the vertex.

7. 7.Which congruence rule proves the converse of the Baudhāyana–Pythagoras theorem?Show answer

   SSS.

8. 8.Name two statements that imply each other.Show answer

   'n is a perfect square' and 'n has an odd number of factors'.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you say why 'Shut the door' is not a proposition?
• ☐Can you rewrite 'Every rhombus is a parallelogram' in if-then form?
• ☐Can you write its converse and decide if it is true?
• ☐Can you give a counterexample to 'every odd number is prime'?
• ☐Can you give a proposition and converse that are both true?
• ☐Can you prove that equal angles give equal opposite sides?
• ☐Can you prove the converse of the Baudhāyana–Pythagoras theorem?
• ☐Can you test whether 20, 21, 29 is a right triangle?

Propositions and their Converses: frequently asked questions