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CLASS 9 MATHEMATICS PART II • CHAPTER 9

Table of Contents

Propositions and Their Converses

Complete Study Notes • Easy Explanations • Solved Examples • Exam Tricks • Practice Quiz

Learn to think logically, identify false statements, and solve mathematical reasoning questions confidently.

Prepared by Grandmaster Bikram Sutradhar

Welcome, Students and Teachers!

Mathematics is not only about calculations. It is also about reasoning, explaining why a statement is true, and finding examples that show when a statement is false.

These original study notes explain the main ideas of Propositions and Their Converses in simple language. Use them for classroom teaching, revision, homework, and examination practice.

Our learning rule: Understand the reason first, practise the method second, and memorise only what is necessary.

1. Learning Objectives

Understand

Recognise mathematical statements that have a definite truth value.

Reverse

Write the converse by swapping the condition and conclusion.

Justify

Explain a true statement using a valid mathematical argument.

Disprove

Find a counterexample that shows a universal claim is false.

2. What Is a Proposition?

Definition: A proposition is a statement that is either true or false, but not both.

Examples of propositions

TRUE

Example 1: 2 + 3 = 5.

The statement has a definite truth value: true.

FALSE

Example 2: 7 is an even number.

The statement has a definite truth value: false.

FALSE

Example 3: Every multiple of 4 is odd.

The statement is false because 8 is a multiple of 4 and is even.

What is not a proposition?

Question: What is your name?
Command: Open your mathematics book.
Unspecified variable: x + 2 = 10, when no value or range for x is given.

Remember: A proposition does not have to be true. A false statement can also be a proposition because it has a definite truth value.

3. What Is a Converse?

Many mathematical propositions are written in the form:

If P, then Q

P is the hypothesis or condition.
Q is the conclusion.

The converse is formed by reversing the two parts.

Original: If P, then Q

Converse: If Q, then P

Worked example

Original statement: If a number is divisible by 6, then it is divisible by 3.

Step 1 — Identify P: The number is divisible by 6.

Step 2 — Identify Q: The number is divisible by 3.

Step 3 — Reverse them: If a number is divisible by 3, then it is divisible by 6.

Important: The original statement is true, but the converse is false. For example, 9 is divisible by 3 but not by 6.

Does a true proposition always have a true converse?

No! The original proposition and its converse must be tested separately.

Original	Converse	Possible relationship
True	True	Both statements hold.
True	False	A true statement may have a false converse.
False	True	The converse can be true even when the original is false.
False	False	Both statements can be false.

4. Counterexamples: The Fastest Way to Disprove a Claim

Definition: A counterexample is a specific case that shows a general statement is false.

If a statement claims that something is true for every number, every shape, or every situation, one valid counterexample is enough to disprove it.

Example 1: Every even number is divisible by 4.

Counterexample: 6 is even, but 6 is not divisible by 4.

Conclusion: The statement is false.

Example 2: Every rectangle is a square.

Consider a rectangle with length 5 cm and width 3 cm.

Its angles are right angles, but its sides are not all equal.

Conclusion: It is a rectangle but not a square.

Example 3: If the road is wet, then it has rained.

A road can become wet because a water tanker spilled water on it.

Conclusion: The converse is not always true.

Exam trick: When asked to disprove a statement, do not write ten examples. Find one correct counterexample and explain why it contradicts the claim.

5. How to Test Whether a Proposition Is True or False

Method A: Proving a statement true

Read the statement carefully.
Identify its condition and conclusion.
Use a mathematical property, definition, or logical argument.
Show that the reasoning works for every allowed case.

Method B: Showing a statement is false

Understand what the statement claims.
Look for a number, shape, or situation that satisfies the condition.
Check whether the conclusion fails for that example.
Write the counterexample and a clear final conclusion.

Common mistake: Checking five examples that work does not prove a statement true for every possible case. A general proof must explain why the result always holds.

6. Important Solved Examples

Example 1: Equality and addition

Statement: If x = y, then a + x = a + y.

Since x = y, adding the same number a to both sides gives:
a + x = a + y.

Converse: If a + x = a + y, then x = y.

Subtract a from both sides:
a + x − a = a + y − a
x = y.

Answer: Both the proposition and its converse are true.

Example 2: Squares of real numbers

Statement: If x = y, then x² = y².

If two numbers are equal, their squares are equal. Therefore, the original statement is true.

Converse: If x² = y², then x = y.

Choose x = 4 and y = −4.

Then x² = 16 and y² = 16, but 4 ≠ −4.

Answer: The converse is false for real numbers. Equality of squares gives x = y or x = −y.

Example 3: Cubes of real numbers

Statement: If x = y, then x³ = y³.

This is true because equal numbers have equal cubes.

Converse: If x³ = y³, then x = y.

The cube function is one-to-one over the real numbers. Therefore, equal cubes imply equal numbers.

Answer: Both directions are true for real numbers.

Example 4: Product of perfect squares

Suppose a = r² and b = s² for integers r and s.

ab = r²s² = (rs)²

Therefore, the product of two perfect squares is a perfect square.

Converse: If ab is a perfect square, then a and b must each be perfect squares.

Take a = 2 and b = 8. Then ab = 16, which is a perfect square, but 2 and 8 are not perfect squares.

Answer: The original statement is true; the converse is false.

Example 5: Divisibility by 24

Statement: If n is divisible by 24, then n is divisible by both 4 and 6.

Since 24 = 4 × 6, every multiple of 24 is divisible by both 4 and 6.

Converse: If n is divisible by both 4 and 6, then n is divisible by 24.

Take n = 12. It is divisible by 4 and 6, but it is not divisible by 24.

Answer: The original is true; the converse is false.

Example 6: Divisibility by 60

If n is divisible by 60, then n is divisible by both 5 and 12.

The original is true because 60 is a multiple of both 5 and 12.

For the converse, 5 and 12 have no common factor greater than 1. Therefore, a number divisible by both must be divisible by their least common multiple:

LCM(5, 12) = 60

Answer: Both the proposition and its converse are true.

7. Number Theory: Tricks and Useful Results

Perfect squares

A positive integer is a perfect square exactly when every exponent in its prime factorisation is even.

Example: 36 = 2² × 3².

Number of factors

If n = pᵃ × qᵇ, where p and q are distinct primes, then the number of positive factors is:

(a + 1)(b + 1)

Divisibility by 3

A number is divisible by 3 exactly when the sum of its digits is divisible by 3.

Example: 246 → 2 + 4 + 6 = 12.

Common factors

For positive integers n and n + 3:

gcd(n, n + 3) = gcd(n, 3)

They have no common factor greater than 1 exactly when n is not divisible by 3.

Factor-counting example

Consider 36 = 2² × 3².

The number of positive factors is:

(2 + 1)(2 + 1) = 9

Its factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

They form partner pairs:

(1, 36), (2, 18), (3, 12), (4, 9), and (6, 6).

The repeated pair (6, 6) explains why a perfect square has an odd number of positive factors.

8. Geometry Connections

Parallel lines and corresponding angles

Proposition: If two lines are parallel, then the corresponding angles formed by a transversal are equal.

This is true by the corresponding-angles property.

Converse: If the corresponding angles formed by a transversal are equal, then the two lines are parallel.

Answer: Both statements are true in the usual Euclidean geometry setting.

Squares and rectangles

Proposition: If a quadrilateral is a square, then all its angles are equal.

A square has four right angles, so the proposition is true.

Converse: If all the angles of a quadrilateral are equal, then it is a square.

A rectangle measuring 5 cm by 3 cm has four equal right angles but is not a square.

Answer: The converse is false. Equal angles alone do not guarantee four equal sides.

Triangle side lengths and opposite angles

In a triangle, equal sides have equal opposite angles, and equal angles have equal opposite sides.

This is a standard example of a result whose two directions can both be proved.

For geometry questions involving a diagram, identify the exact points, angle bisectors, and line segments before writing a proof. Do not assume that a diagram alone proves a relationship.

The Baudhāyana–Pythagoras theorem and its converse

Theorem: If a triangle is right-angled, then the squares of its two perpendicular sides add up to the square of its hypotenuse.

a² + b² = c²

Converse: If the side lengths of a triangle satisfy a² + b² = c², then the triangle is right-angled.

The converse is also true. It is an important example of a proposition and its converse both being true.

9. Exercise Set 9.1 — Revision Guide for Questions 1–17

The following is an original revision guide based on the chapter topics. It summarises the key reasoning needed for the exercise. Use the precise wording and diagrams in your textbook when writing final answers.

Question 1 — Parallel lines

Key result: Parallel lines give equal corresponding angles. Equal corresponding angles also establish that the lines are parallel.

Conclusion: Both directions are true.

Question 2 — Square and equal angles

Key result: Every square has four equal angles.

Counterexample to the converse: A non-square rectangle.

Conclusion: Original true; converse false.

Question 3 — Triangle angle bisectors

This question uses the figure involving triangle ABC, its incenter I, and the extended angle bisectors meeting the opposite sides at E and F.

Exam method: Mark the given equality AB = AC, identify the symmetry of the isosceles triangle, and explain how that symmetry relates IE and IF. For the converse, start from IE = IF and establish the required side relationship using the geometry of the construction.

This is a figure-dependent question. Copy the construction accurately and show the intermediate angle or triangle relationships required by your textbook; do not replace the proof with “the diagram looks equal.”

Question 4 — Adding the same number

If x = y, then a + x = a + y. Conversely, if a + x = a + y, subtract a from both sides to obtain x = y.

Conclusion: Both directions are true.

Question 5 — Product of perfect squares

Write a = r² and b = s². Then ab = (rs)², a perfect square.

Converse counterexample: 2 × 8 = 16, but neither 2 nor 8 is a perfect square.

Question 6 — Equality of squares

If x = y, then x² = y².

Counterexample to the converse: x = 4 and y = −4. Their squares are equal, but the numbers are not.

Question 7 — Equality of cubes

If x = y, then x³ = y³. Conversely, equal real cubes imply equal real numbers.

Conclusion: Both directions are true for real numbers.

Question 8 — Divisibility by 24

A multiple of 24 is divisible by 4 and 6.

Counterexample to the converse: 12 is divisible by 4 and 6 but not by 24.

Question 9 — Divisibility by 60

Since LCM(5, 12) = 60, a number divisible by both 5 and 12 must be divisible by 60.

Conclusion: Both directions are true.

Question 10 — Square of a prime

If n = p² for a prime p, its positive factors are 1, p and p².

Converse idea: A positive integer with exactly three positive factors must be the square of a prime.

Question 11 — Product of two unequal primes

If n = pq for distinct primes p and q, its positive divisors are 1, p, q and pq.

Converse counterexample: 8 has four positive divisors, but it is not a product of two unequal primes.

Question 12 — Common factors of n and n + 3

Use gcd(n, n + 3) = gcd(n, 3).

Conclusion: The two numbers have no common factor greater than 1 exactly when n is not a multiple of 3.

Question 13 — Finding counterexamples to prime claims

(i) For expressions of the form 4n² + 1, take n = 4:

4(4²) + 1 = 65 = 5 × 13

(ii) For expressions of the form n² + n + 11, take n = 10:

10² + 10 + 11 = 121 = 11 × 11

(iii) For expressions of the form 4ⁿ + 3, take n = 4:

4⁴ + 3 = 259 = 7 × 37

Conclusion: Each composite value disproves the claim that every value of the expression is prime.

Question 14 — More counterexamples

(i) For the claim that 2ⁿ − 1 is prime whenever n is prime, take n = 11:

2¹¹ − 1 = 2047 = 23 × 89

(ii) For the claim that 2ⁿ + 1 is prime whenever n is even, take n = 6:

2⁶ + 1 = 65 = 5 × 13

Conclusion: Both claims have counterexamples.

Question 15 — Divisibility by 8

If a number is divisible by 8, it is divisible by both 2 and 4 because 8 is a multiple of 2 and 4.

However, checking divisibility by 2 and 4 is not enough to establish divisibility by 8.

Counterexample: 12 is divisible by 2 and 4 but not by 8.

Question 16 — Divisibility by 3 and digit sums

Direction 1: If a number is divisible by 3, then the sum of its digits is divisible by 3.

Direction 2: If the sum of a number’s digits is divisible by 3, then the number is divisible by 3.

Conclusion: Both directions are true.

Question 17 — Constructing quadrilaterals using two sticks

(i) If every quadrilateral of type Q has equal-length diagonals:

(a) The sticks should be equal in length because the diagonals must have equal lengths.

(b) The way the sticks are joined can matter. Equal diagonals are a necessary property, but that property alone does not guarantee that every arrangement produces a quadrilateral of type Q.

(ii) If every quadrilateral with equal diagonals is of type Q:

(a) The sticks must be equal in length to meet the condition of equal diagonals.

(b) Under the stated assumption, the arrangement does not affect the conclusion: any permitted construction with equal diagonals must be of type Q.

Always distinguish between “Q implies equal diagonals” and “equal diagonals imply Q.” These are converse statements, and their construction requirements are different.

10. Extra Exam Practice with Solutions

Try answering each question before opening its solution.

Practice 1 — Identify the converse

If a number is divisible by 10, then it is divisible by 5.

Converse: If a number is divisible by 5, then it is divisible by 10.
Truth: False.
Counterexample: 15 is divisible by 5 but not by 10.

Practice 2 — Test the converse

If a triangle is equilateral, then all its sides are equal.

Converse: If all three sides of a triangle are equal, then the triangle is equilateral.
Truth: Both statements are true by the definition of an equilateral triangle.

Practice 3 — Find a counterexample

Every number divisible by 6 is divisible by 12.

Counterexample: 18 is divisible by 6 but not by 12.
Therefore, the statement is false.

Practice 4 — Perfect squares

If a number is a perfect square, then it has an odd number of positive factors.

Truth: True.
Factors normally pair as f and n/f. For a perfect square, one factor is paired with itself at the square root. This gives an odd total number of positive factors.

Practice 5 — Divisibility by 3

Is 5,472 divisible by 3?

Add the digits: 5 + 4 + 7 + 2 = 18.
Since 18 is divisible by 3, 5,472 is divisible by 3.

Exam Answer-Writing Formula

Original statement → Converse → Truth value → Proof or counterexample → Conclusion

Following this order makes your reasoning easy for teachers and examiners to follow.

11. Interactive Mathematics Quiz

Test your understanding. Select one answer for each question.

1. What is the converse of “If P, then Q”?

2. Which is a counterexample to “Every even number is divisible by 4”?

3. If x = y, which statement must be true?

4. A rectangle that is not a square is a counterexample to which claim?

5. What is enough to disprove a universal mathematical claim?

12. Last-Minute Exam Tricks

Trick 1: Reverse carefully

Swap the complete condition and conclusion. Do not change only one word.

Trick 2: Test separately

Never assume that the converse is true just because the original is true.

Trick 3: Use small numbers

For divisibility claims, test simple numbers such as 6, 8, 12, 15 and 18.

Trick 4: Explain the failure

After giving a counterexample, explicitly say which condition holds and which conclusion fails.

Trick 5: Show your proof

Write the property used at each step instead of giving only the final answer.

Trick 6: Check the domain

Notice whether the question refers to natural numbers, positive integers, or all real numbers.

13. Seven-Day Practice Plan

Day	Topic	Daily task
Day 1	Propositions	Classify 15 statements as propositions or non-propositions.
Day 2	Converses	Write the converse of 15 if-then statements.
Day 3	Counterexamples	Disprove 10 false claims using suitable examples.
Day 4	Divisibility	Practise questions involving 3, 4, 6, 8, 12 and 24.
Day 5	Geometry	Review squares, rectangles, triangles and parallel lines.
Day 6	Textbook exercise	Attempt Exercise Set 9.1 without looking at the solutions.
Day 7	Mock test	Complete a timed test and correct every mistake.

14. Frequently Asked Questions

Q1. Is every true proposition’s converse also true?

No. For example, if a number is divisible by 6, it is divisible by 3. The converse fails for 9.

Q2. What is the easiest way to disprove a statement?

Find one valid counterexample that satisfies the condition but makes the conclusion false.

Q3. Does a false proposition have to be a non-proposition?

No. A false statement with a definite truth value is still a proposition.

Q4. How should I answer a converse question in an exam?

Write the converse first, determine its truth value, and then provide a proof or counterexample.

Q5. Is this lesson a replacement for the textbook?

No. Use these notes for revision and explanation, and consult your prescribed textbook for the exact exercise wording and diagrams.

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Final Message to Students

A good mathematician does not simply memorise an answer. A good mathematician asks: “Why is this true?” and “Can I find a case where it fails?”

Practise a little every day, write complete reasoning, and learn from mistakes. With consistent effort, difficult mathematical ideas become easier to understand.

Keep learning and keep improving!

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