Propositions and their Converses: Class 9 Ganita Manjari · Chapter 9

Ganita Manjari Part 2 · Companion notes, simulations and practice · Official NCERT chapter PDF ↗

Published 13 Oct 2026 · Updated 13 Oct 2026

45 min
22 practice questions
17 MCQs

What you will learn in Propositions and their Converses

Recognise a proposition as a sentence that is either true or false, and write it in the form 'if X then Y'
Form the converse of a proposition by swapping its 'if' and 'then' parts, keeping any setting unchanged
Decide whether a proposition and its converse are true, proving the true ones and giving a counterexample for each false one
Explain why one counterexample disproves a claim while many agreeing examples do not prove it
Prove the converses of the isosceles-triangle theorem and of the Baudhāyana–Pythagoras theorem
Recognise the same meaning in other phrasings such as 'X implies Y' and 'Y when X'

Original companion text aligned to NCERT Mathematics Class 9, Chapter 9, pp. 1–7. Not affiliated with or endorsed by NCERT.

9.1 Propositions and the 'if … then …' form

Some sentences can be checked and some cannot. 'Kolkata lies on the Hooghly' is true. ' $17$ is an even number' is false. But 'Please pass the salt' is a request, 'Is the bus late?' is a question and 'Biryani is the best food' is an opinion. None of these three can be called simply true or false.

Definition

Proposition

A sentence that is either true or false, but not both.

Example' $91$ is a prime number' is a proposition. It happens to be false, because $91 = 7 \times 13.$

So a false sentence still counts as a proposition (कथन). What matters is that it has a definite answer, true or false. Many propositions in mathematics join two smaller statements with the words if and then. Here is one you proved in an earlier grade.

TheoremAngles facing equal sides

In a triangle, if two sides have the same length, then the angles facing those two sides are also equal.

Call the part after if $X$ , and the part after then $Y$ . Here $X$ is 'two sides are equal' and $Y$ is 'the angles facing them are equal'. Now swap the two parts. You get a new proposition: in a triangle, if two angles are equal, then the sides facing those angles have the same length.

Definition

Converse

The converse of 'if $X$ then $Y$ ' is the proposition 'if $Y$ then $X$ '. You make it by swapping the 'if' part and the 'then' part.

ExampleThe converse of 'if a number ends in 5, then it is odd' is 'if a number is odd, then it ends in 5'.

3/3A banana is a fruit but not a mango, so the converse is false.

Fig. 9.1Swapping the 'if' card and the 'then' card turns a proposition into its converse.

Two slots read 'If' and 'then'. In the first step the orange card X, 'it is a mango', sits in the If slot and the green card Y, 'it is a fruit', in the then slot; a panel says True: every mango is a fruit. In the second step the two cards trade places, so the sentence reads 'If it is a fruit, then it is a mango', and the panel asks whether every fruit is a mango. In the third step the panel shows a cross, the word False and a banana: a banana is a fruit but not a mango.

We often write 'if $X$ then $Y$ ' as ' $X$ implies $Y$ ', or with an arrow: $X \Rightarrow Y .$ Read the arrow as 'implies'. The converse (विलोम) is then $Y \Rightarrow X .$ The arrow simply points the other way.

In Fig. 9.1 a true proposition produced a false converse. So the big question of this chapter is: does truth survive the swap? Sometimes it does and sometimes it does not, as the next sections show.

Quick check

1 of 3

Which of these are propositions? Pick all that apply.

Select all that apply

Show answer

Answer

(A) $17$ is a prime number.
(C) Every square has four equal sides.
(E) $5 + 3 = 9$

A and C are true propositions and E is a false one: a false sentence is still a proposition. B is a request and D is a question, so neither can be true or false.

9.2 Is the converse of a true proposition true?

Start with the pair of triangle propositions from Section 9.1. The first one is a theorem you already know. Before reading on, try to settle the second one yourself.

3/4 $△ P Q S$ and $△ P R S$ match angle, angle and side.

Fig. 9.2Dropping the altitude splits the triangle into two congruent halves.

Triangle PQR has its apex P at the top and equal angles marked at Q and R; the note asks whether PQ equals PR. In the second step a dashed altitude PS drops from P to QR with right-angle marks at S. In the third step the two halves, PQS and PRS, are shaded in different colours and the notes list the matching parts: both angles at S are 90 degrees, PS is shared, so the triangles are congruent. In the last step tick marks appear on PQ and PR and the note concludes PQ = PR.

TheoremSides facing equal angles

If a triangle has two equal angles, then the sides facing those angles are equal.

Let $∠ Q = ∠ R$ in $△ P Q R .$ Draw $P S$ perpendicular to $Q R$ , meeting it at $S$ (Fig. 9.2).

In $△ P Q S$ and $△ P R S,$ two pairs of angles are equal: $∠ Q = ∠ R$ (given) and $∠ P S Q = ∠ P S R = 90^{\circ} .$ The angles of a triangle add up to $180^{\circ}$ , so the third pair is equal too: $∠ Q P S = ∠ R P S .$

The side $P S$ belongs to both triangles and lies between the matching angles at $S$ and at $P$ . So the triangles are congruent by ASA, and their matching sides are equal: $P Q = P R$ .

So here the proposition and its converse are both true. Does that always happen? Try an everyday proposition.

Example 1NCERT Example 1 · p. 1

A city and its state

Proposition P: if a person lives in Pune, then the person lives in Maharashtra. Write its converse Q. Is P true? Is Q true?

Try it yourself first. Then reveal the solution one step at a time.

3/3A Nagpur resident is in Y but not in X: a counterexample to Q.

Fig. 9.3'If X then Y' says the X region sits inside the Y region. A point in Y outside X breaks the converse.

A large blue oval stands for everyone who lives in Maharashtra (Y) and a smaller orange circle inside it for everyone who lives in Pune (X). In the first step only the two regions are shown, the X circle sitting entirely inside the Y oval. In the second step a dot for Meera, who lives in Pune, appears inside the circle with a tick: being in X puts her in Y. In the third step a diamond for a Nagpur resident appears inside the oval but outside the circle, with a cross and the note 'in Y but not in X: counterexample to the converse'.

The Nagpur resident is a counterexample to Q. In a picture like Fig. 9.3, a counterexample to 'if $Y$ then $X$ ' is a point inside the $Y$ region that lies outside the $X$ region.

Definition

Counterexample

One particular case that makes the 'if' part of a proposition true and its 'then' part false. A single counterexample proves that the proposition is false.

Example $9$ is a counterexample to 'if a number is odd, then it is prime': $9$ is odd, but $9 = 3 \times 3.$

Proving and disproving are very unequal jobs. A claim of truth must be backed by an argument that works for every case, even cases nobody has checked. A claim of falsehood needs just one case (प्रत्युदाहरण), no matter how many other cases agreed. That is why mathematicians hunt for counterexamples.

From historyNCERT p. 2

Five primes, then a surprise

Around 1640, Pierre de Fermat studied numbers of the form $2^{2^{n}} + 1.$ For $n = 0, 1, 2, 3, 4$ every one of them is prime, and he believed the pattern would never break.

Table 9.1The first six numbers of the form

2^{2^{n}} + 1

$n$	$2^{2^{n}} + 1$	Prime?
$0$	$3$	yes
$1$	$5$	yes
$2$	$17$	yes
$3$	$257$	yes
$4$	$65 537$	yes
$5$	$4 294 967 297$	no

In 1732 Leonhard Euler showed that $2^{32} + 1 = 641 \times 6 700 417.$ Five agreeing cases could not rescue the claim; one counterexample ended it. No other prime of this form has been found since.

Example 2NCERT Example 2 · p. 2

Multiples of 12 and of 4

Here is a claim P: if a whole number is divisible by 12, then 4 divides it too. Explain why P holds. Then write its converse Q and decide whether Q holds.

Try it yourself first. Then reveal the solution one step at a time.

9.3 When both directions are true

Sometimes the converse survives every test. A classic case comes from counting factors. List the factors of a few numbers and count them: which numbers end up with an odd count?

Example 3NCERT Example 3 · p. 2

Perfect squares and odd factor counts

Take any positive integer $n$ . Claim P: if $n = m \times m$ for some whole number $m$ , then the count of factors of $n$ is odd. Its converse Q: if the count of factors of $n$ is odd, then $n = m \times m$ for some whole number $m$ . Test both claims on $40$ and on $36$ .

Try it yourself first. Then reveal the solution one step at a time.

3/3 $6$ pairs with itself, since $6 \times 6 = 36 :$ the count is odd.

Fig. 9.4Factors come in pairs. Only a square has a factor that pairs with itself.

Two rows of numbers. The top row lists the factors of 40: 1, 2, 4, 5, 8, 10, 20, 40. In the first step blue arcs join 1 with 40, 2 with 20, 4 with 10 and 5 with 8: four pairs, eight factors. The bottom row lists the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. In the second step green arcs join 1 with 36, 2 with 18, 3 with 12 and 4 with 9, leaving 6 in the middle with no arc. In the third step the 6 is ringed with a dashed loop and a note says 6 × 6 = 36, so 36 has 9 factors.

Which direction does an argument prove?

Dividing $n$ by any factor $f$ gives another factor, $n \div f$ , and the two multiply to $n$ . In Fig. 9.4 each arc joins such a factor pair. An arc between two different numbers covers two factors, so arcs alone always account for an even number of factors.

Now suppose the count of factors is odd. Arcs cannot cover everything, so some factor $f$ has no arc: it is its own mate, with $f \times f = n .$ Then $n$ is a perfect square. This argument starts from odd count and ends at square, so what it proves is Q. It tells us nothing yet about P.

To prove P, start from the other end. If $n = f \times f,$ then $f$ is paired with itself. Could a second number $g$ also be paired with itself? Then $g \times g = f \times f .$ But if $g > f$ then $g \times g > f \times f,$ and if $g < f$ then $g \times g < f \times f .$ So $g = f$ : a number has at most one self-paired factor.

All the other factors sit in pairs of two different numbers, which give an even count. Adding the one self-paired factor makes the total odd. So P is true as well.

Quick check

1 of 3

How many factors does

100

have?

factors

Show answer

Answer 9 factors

1 \times 100,

2 \times 50,

4 \times 25,

5 \times 20

and

10 \times 10 :

four pairs plus the self-paired

10

make

9

factors, an odd count, as P predicts for the square

100

9.4 Four possible outcomes

So far we have seen a true proposition with a false converse, and pairs where both directions hold. Can the proposition be the false one while its converse is true?

Example 4NCERT Example 4 · p. 3

Equal perimeters and congruent triangles

P: if two triangles have equal perimeters, then they are congruent. Q: if one triangle is congruent to another, then the two perimeters match. Decide whether each is true. Justify the true one and give a counterexample for the false one.

Try it yourself first. Then reveal the solution one step at a time.

3/3B does not fit on A, so P is false.

Fig. 9.5Equal perimeters do not force congruence.

In the first step a blue right-angled triangle A with sides 6, 8 and 10 cm is shown with its perimeter, 6 + 8 + 10 = 24 cm. In the second step an orange equilateral triangle B with sides of 8 cm appears beside it, also with perimeter 24 cm. In the third step a dashed copy of triangle B slides on top of triangle A: the outlines do not match, and the note reads 'Same perimeter, not congruent'.

Put the examples side by side and every combination turns up (Table 9.2).

Table 9.2A proposition and its converse: all four combinations happen

	Example
Both true	perfect square $\Rightarrow$ odd number of factors (Example 3)
Only the proposition true	multiple of 12 $\Rightarrow$ multiple of 4 (Example 2)
Only the converse true	equal perimeters $\Rightarrow$ congruent triangles (Example 4)
Both false	prime $\Rightarrow$ odd (2 is an even prime; 9 is odd but not prime)

The lesson: knowing whether a proposition is true tells you nothing about its converse. Each one must be checked on its own, with a proof or with a counterexample. The checker below lets you test some number propositions and their converses on the numbers 1 to 100.

InteractiveConverse checker

A menu offers six number propositions, such as 'multiple of 12 implies multiple of 4' and 'prime implies odd'. Two buttons choose between the proposition and its converse, and the sentence is written out in full. Pressing 'Check 1 to 100' marks a 10 by 10 grid of the numbers 1 to 100: numbers that make the 'if' part true get a dark outline, numbers that make the 'then' part true are shaded, and any number that makes the 'if' part true and the 'then' part false gets a thick orange border and a small cross as a counterexample. A status line lists the first counterexamples, or says that none was found up to 100 and that this is evidence, not a proof.

Fig. 9.6Pick a proposition, choose it or its converse, then check every number from 1 to 100 for counterexamples.

Quick check

1 of 2

Match each proposition with what is true about it and its converse.

Tap an item on the left, then its match on the right.

Show answer

Answer

multiple of 8 $\Rightarrow$ even : only the proposition
three equal sides $\Rightarrow$ three equal angles : both
even $\Rightarrow$ multiple of 4 : only the converse
prime $\Rightarrow$ odd : neither

8 k = 2 \times 4 k

is even, but

6

is even and not a multiple of 8. A triangle has three equal sides exactly when it has three equal angles.

6

is even but not a multiple of 4, while every multiple of 4 is even.

2

is an even prime, and

9

is odd but not prime.

9.5 The Baudhāyana–Pythagoras theorem and its converse

You have used the Baudhāyana–Pythagoras theorem many times, for example to find distances between points in Chapter 1 of Part 1. Here is a careful way to state it.

TheoremBaudhāyana–Pythagoras theorem

In a triangle with sides $a$ , $b$ and $c$ : if the angle between the sides $a$ and $b$ is a right angle, then $a^{2} + b^{2} = c^{2} .$

Example 5NCERT Example 5 · p. 4

Writing the converse of the theorem

Write the converse of the Baudhāyana–Pythagoras theorem as stated above.

Try it yourself first. Then reveal the solution one step at a time.

Writing a converse correctly is only half the job: it still has to be proved. The proof below uses the theorem itself, applied to a second triangle that we build.

4/4Same three sides, so $∠ R = ∠ F = 90^{\circ} .$

Fig. 9.7Build a right triangle with legs

a

and

b

; its hypotenuse turns out to be

c

On the left, triangle PQR lies on its longest side PQ = c, with QR = a, PR = b and a question mark at the top vertex R; the label above says a squared plus b squared equals c squared. In the second step a right triangle DEF appears on the right, with the right angle at F, EF = a and DF = b. In the third step its third side DE is labelled c, because DE squared equals a squared plus b squared, which equals c squared. In the last step the question mark at R is replaced by a right-angle mark: the triangles are congruent by SSS, so angle R equals angle F, 90 degrees.

TheoremConverse of the Baudhāyana–Pythagoras theorem

If the sides of a triangle satisfy $a^{2} + b^{2} = c^{2},$ then the angle opposite the side $c$ is a right angle.

Take $△ P Q R$ with $Q R = a$ , $P R = b$ , $P Q = c$ and $a^{2} + b^{2} = c^{2} .$ We must show that $∠ R = 90^{\circ} .$

Build a second triangle $D E F$ with a right angle at $F$ , $E F = a$ and $D F = b$ (Fig. 9.7).

$△ D E F$ is right-angled, so the theorem applies to it: $D E^{2} = a^{2} + b^{2} .$ But $a^{2} + b^{2} = c^{2},$ so $D E^{2} = c^{2}$ . Lengths are positive, so $D E = c$ .

Now both triangles have sides $a$ , $b$ and $c$ , so they are congruent by SSS. The angle at $R$ faces the side $c$ , and so does the angle at $F$ . Matching angles are equal, so $∠ R = ∠ F = 90^{\circ} .$

Notice that the proof uses the original theorem only on $△ D E F,$ which is right-angled because we built it that way. It never assumes anything about $∠ R$ , so the argument is not going round in a circle.

Example

Is there a right angle?

Which of these triangles has a right angle, and where? (a) Sides 20 cm, 21 cm and 29 cm. (b) Sides 7 cm, 8 cm and 11 cm.

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 3

A triangle has sides 9 cm, 40 cm and 41 cm. Which statement is true?

It has no right angle, because

9 + 40 \neq 41.

It has a right angle facing the 41 cm side.It has a right angle facing the 9 cm side.We cannot tell without measuring an angle.

Show answer

Answer (B) It has a right angle facing the 41 cm side.

9^{2} + 40^{2} = 81 + 1600 = 1681 = 41^{2},

so by the converse the angle facing the 41 cm side is

90^{\circ}

. Adding the sides, as in the first option, is not the test.

9.6 Other ways to say 'if X then Y'

Everyday English has many ways to express the same 'if … then …' idea. Every sentence in Table 9.3 says exactly the same thing as if a number is a multiple of 10, then it is even.

Table 9.3Six ways to say 'if X then Y'NCERT A Note · p. 5

Pattern	Example
If $X$ , then $Y$ .	If a number is a multiple of 10, then it is even.
$X$ implies $Y$ ( $X \Rightarrow Y$ ).	Being a multiple of 10 implies being even.
$Y$ when $X$ .	A number is even when it is a multiple of 10.
$Y$ whenever $X$ .	A number is even whenever it is a multiple of 10.
Every $X$ is $Y$ .	Every multiple of 10 is even.
$X$ only if $Y$ .	A number is a multiple of 10 only if it is even.

4/4A number is a multiple of 10 only if it is even. Still $X \Rightarrow Y .$

Fig. 9.8The tiles move around the sentence, but

X

is always the condition and

Y

the result.

Two tiles, an orange X tile reading 'a multiple of 10' and a green Y tile reading 'even', are placed into four sentence patterns in turn. Step 1: 'If X, then Y'. Step 2: the tiles move to read 'Y when X'. Step 3: 'Every X is Y'. Step 4: 'X only if Y'. Under every pattern the same line reads X implies Y, showing that the meaning never changes even though the tiles move around the sentence.

The last pattern trips many people up. ' $X$ only if $Y$ ' does not mean $Y \Rightarrow X .$ It says that $X$ never happens without $Y$ , which is $X \Rightarrow Y .$ So before you form a converse, rewrite the sentence in 'if … then …' form, pick out $X$ and $Y$ , and only then swap them.

Try thisNCERT Think and Reflect · p. 5

Converse hunt

Find three true propositions whose converses are false: one from daily life, one from geometry and one about numbers. For each, write the converse and a counterexample to it.

Table 9.4My converse hunt

	Proposition (if … then …)	Converse	Counterexample to the converse
Daily life
Geometry
Numbers

Your entries are saved automatically.

Recap

Propositions and their Converses: key points and summary

A proposition is a sentence that is either true or false (Section 9.1).
The converse of 'if $X$ then $Y$ ' is 'if $Y$ then $X$ '. A scene-setting phrase such as 'in a triangle with sides $a$ , $b$ , $c$ ' stays unchanged (Example 5).
' $X$ implies $Y$ ', ' $X \Rightarrow Y$ ', ' $Y$ when $X$ ' and 'every $X$ is $Y$ ' all mean 'if $X$ then $Y$ ' (Section 9.6).
A counterexample makes the 'if' part true and the 'then' part false. One is enough to show a proposition is false (Section 9.2).
Many agreeing cases do not make a proof: Fermat's numbers were prime five times and then failed (the Fermat story).
A proposition and its converse can both be true, both be false, or have one of each (Table 9.2).
A number is a perfect square if and only if it has an odd number of factors. Each direction needs its own argument (Section 9.3).
In a triangle, equal angles face equal sides, and $a^{2} + b^{2} = c^{2}$ forces a right angle facing $c$ : both converses are true (isosceles converse, Baudhāyana–Pythagoras converse).

Exercise Set 9.1

Find your NCERT exercise question

Our practice questions are written for ncert.ai and follow the NCERT exercises. Find the one that matches the question in your textbook; its answer is with it.

In your NCERT textbook	Practice question here
NCERT Exercise Set 9.1, Q1	Practice Q1
NCERT Exercise Set 9.1, Q2	Practice Q2
NCERT Exercise Set 9.1, Q3	Practice Q3
NCERT Exercise Set 9.1, Q4	Practice Q4
NCERT Exercise Set 9.1, Q5	Practice Q5
NCERT Exercise Set 9.1, Q6	Practice Q6
NCERT Exercise Set 9.1, Q7	Practice Q7
NCERT Exercise Set 9.1, Q8	Practice Q8
NCERT Exercise Set 9.1, Q9	Practice Q9
NCERT Exercise Set 9.1, Q10	Practice Q10
NCERT Exercise Set 9.1, Q11	Practice Q11
NCERT Exercise Set 9.1, Q12	Practice Q12
NCERT Exercise Set 9.1, Q13	Practice Q13
NCERT Exercise Set 9.1, Q14	Practice Q14
NCERT Exercise Set 9.1, Q15	Practice Q15
NCERT Exercise Set 9.1, Q16	Practice Q16
NCERT Exercise Set 9.1, Q17	Practice Q17

Quick check: MCQs on Propositions and their Converses

9.1 Propositions and the 'if … then …' form3 questions
9.2 Is the converse of a true proposition true?3 questions
Which direction does an argument prove?3 questions
9.4 Four possible outcomes2 questions
9.5 The Baudhāyana–Pythagoras theorem and its converse3 questions
9.6 Other ways to say 'if X then Y'3 questions

Glossary · 8

Key terms (English–Hindi)

congruentसर्वांगसमsarvaangsam: Two figures are congruent when one can be placed exactly on the other: they have the same shape and the same size.First seen in the chapter →
converseविलोमvilom: The proposition you get by swapping the 'if' part and the 'then' part: the converse of 'if $X$ then $Y$ ' is 'if $Y$ then $X$ '.First seen in the chapter →
counterexampleप्रत्युदाहरणpratyudaharan: One particular case that makes the 'if' part of a proposition true and its 'then' part false. One counterexample is enough to show the proposition is false.First seen in the chapter →

factor pair: Two whole numbers whose product is a given number, such as $4$ and $9$ for $36$ . A pair may use one number twice, as in $6 \times 6$ .First seen in the chapter →

if and only ifयदि और केवल यदिyadi aur keval yadi: ' $X$ if and only if $Y$ ', written $X \Leftrightarrow Y,$ says that a proposition and its converse are both true.First seen in the chapter →
implies: ' $X$ implies $Y$ ', written $X \Rightarrow Y,$ means the same as 'if $X$ then $Y$ ': whenever $X$ is true, $Y$ is true too.First seen in the chapter →

perfect squareपूर्ण वर्गpoorn varg: A whole number that equals some whole number multiplied by itself, such as $49 = 7 \times 7.$ First seen in the chapter →
propositionकथनkathan: A sentence that is either true or false, but not both. A false sentence such as ' $15$ is a multiple of $4$ ' is still a proposition.First seen in the chapter →

Propositions and their Converses: Class 9 Ganita Manjari · Chapter 9

What you will learn in Propositions and their Converses

9.1 Propositions and the 'if … then …' form

9.2 Is the converse of a true proposition true?

9.3 When both directions are true

Which direction does an argument prove?

9.4 Four possible outcomes

9.5 The Baudhāyana–Pythagoras theorem and its converse

9.6 Other ways to say 'if X then Y'

Propositions and their Converses: key points and summary

Exercise Set 9.1

Practice questions with answers (NCERT “Exercise Set 9.1”)

Find your NCERT exercise question

Quick check: MCQs on Propositions and their Converses

Key terms (English–Hindi)

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