Free Class 9 Maths Ch.9 PPT (81 slides) - Propositions and their Converses, Ganita Manjari Part II. Converses, counterexamples, Exercise 9.1. NCERT 2026-27.
This free PPT Slides for CBSE Class IX Maths, Chapter 9: Propositions and their Converses, contains a chapter-wise PowerPoint presentation with visual slides, diagrams and key points for classroom and self-study. It has been prepared by Sumeet Sahu at Unique Study Point, Indore, strictly following the latest NCERT syllabus for Session 2026-27.
Free notes, PPT, and worksheets for Class 9 Mathematics Chapter 9 from Ganita Manjari Part II, based on the new NCERT syllabus (Session 2026–27). This chapter introduces logical reasoning in mathematics — covering Propositions and Converses, Counterexamples, When Both Ways Hold, the Converse of the Baudhayana–Pythagoras Theorem, and Exercise 9.1.
A proposition is a mathematical statement that is either true or false — never both. For example, "If a number ends in 0, then it is divisible by 5" is a proposition. Most mathematical propositions take an "if–then" form, where the "if" part is called the hypothesis and the "then" part is called the conclusion.
The converse of a proposition is formed by swapping its hypothesis and conclusion. If the original statement is "If P, then Q", its converse is "If Q, then P".
A very important idea in this chapter: a proposition being true does not mean its converse is also true. Each has to be checked separately.
| Statement | Converse | Both true? |
|---|---|---|
| If a number ends in 0, it is divisible by 5 | If a number is divisible by 5, it ends in 0 | No — 15 is divisible by 5 but ends in 5 |
| If a shape is a square, it is a rectangle | If a shape is a rectangle, it is a square | No — a 4×2 rectangle is not a square |
| If a triangle is equilateral, all its angles are 60° | If all angles are 60°, the triangle is equilateral | Yes — both are true |
A counterexample is a single example that proves a statement false. To disprove any general claim, you do not need many examples — just one that breaks the rule is enough.
For instance, to disprove "every odd number is prime", the number 9 is enough: it is odd, but 9 = 3 × 3, so it is not prime. That one example settles it.
Sometimes a proposition and its converse are both true. In such cases, mathematicians say the two statements are equivalent, and write them together using "if and only if".
Example: "A triangle is equilateral if and only if all its angles measure 60°." Here both directions hold, so the two conditions always occur together.
The Baudhayana–Pythagoras Theorem states: in a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides (a² + b² = c²).
Its converse states: if in a triangle the square of one side equals the sum of the squares of the other two sides, then the triangle is right-angled. Both the theorem and its converse are true — and the converse is what lets us test whether a triangle is right-angled just from its side lengths.
Example: a triangle with sides 3, 4 and 5 satisfies 3² + 4² = 9 + 16 = 25 = 5², so by the converse it must be a right-angled triangle.
The converse of the Baudhayana–Pythagoras theorem is used by carpenters and masons to check whether a corner is a perfect right angle — the well-known "3-4-5 method" used while laying foundations and fitting door frames. Logical reasoning with propositions and counterexamples is also the foundation of computer programming, legal reasoning, and scientific proof.
Q1. What is the converse of a statement?
The converse is formed by interchanging the hypothesis and the conclusion. The converse of "If P, then Q" is "If Q, then P".
Q2. If a statement is true, is its converse always true?
No. A statement and its converse are independent — one can be true while the other is false. Each must be verified separately.
Q3. What is a counterexample?
A counterexample is a single case that shows a general statement is false. Just one counterexample is enough to disprove a claim, no matter how many examples support it.
Q4. What does "if and only if" mean?
It means both a statement and its converse are true, so the two conditions always occur together — each one implies the other.
Q5. How is the converse of the Baudhayana–Pythagoras theorem used?
It lets you identify a right-angled triangle from its side lengths alone. If a² + b² = c², the triangle must be right-angled — which is how the 3-4-5 check works in construction.
Prepared by Sumeet Sahu, Unique Study Point, Indore. Free for all Class 9 CBSE students.
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| Class | Class IX (CBSE / NCERT) |
| Subject | Maths |
| Chapter | Chapter 9: Propositions and their Converses |
| Resource Type | PPT Slides |
| Last Updated | 29 September 2026 |
| Session | 2026-27 (Latest NCERT Syllabus) |
| Downloads | 159+ |
| Prepared by | Sumeet Sahu, Unique Study Point, Indore |
| Cost | Free |