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Class 9 Maths: Propositions and their Converses, 8 practice questions with answers

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These are 8 practice questions on Chapter 9, Propositions and their Converses, of the new NCERT Class 9 Maths textbook, Ganita Manjari, Part II (First Edition, August 2026). Each has a worked answer and the textbook page it comes from. They cover telling a proposition from a sentence that is not one, writing a converse and deciding whether it is true, finding a counterexample, a pair of statements that imply each other, a pair that are both false, and rewriting a "when" sentence in if-then form. Try each one on paper before you open the answer.

These questions are our own, drafted with AI help. Every number was re-computed by a test script and none of the numbers is taken from the chapter, but no practising teacher has reviewed them yet. If you find a mistake, please tell us.

What the chapter says you need

Facts from the NCERT textbook (Ganita Manjari, Grade 9, Part II, Chapter 9, printed pages 1 to 7):

Propositions and converses

Question 1. Which of these sentences are propositions? For each proposition, say whether it is true or false. (a) Close the door. (b) 53 is a prime number. (c) Every triangle has four sides. (d) Is 91 a prime number?

Answer: (b) and (c) are propositions; (b) is true and (c) is false. (a) and (d) are not propositions. A proposition must be either true or false. "53 is a prime number" is true, because 53 has no factor other than 1 and itself. "Every triangle has four sides" is false. An instruction such as (a) and a question such as (d) are neither true nor false, so they are not propositions. (Proposition, page 1.)

Question 2. Proposition: "If a triangle is equilateral, then it is isosceles." Is it true? Write its converse and decide whether the converse is true.

Answer: the proposition is true; the converse, "If a triangle is isosceles, then it is equilateral", is false. An isosceles triangle has at least two equal sides; an equilateral triangle has all three sides equal, so it certainly has two equal sides and the proposition holds. For the converse, one counterexample is enough: an isosceles triangle whose two equal sides are longer than its third side is isosceles but not equilateral. (Converse and counterexample, page 1.)

Question 3. Proposition: "If an animal is a cow, then it has four legs." Write the converse. Which of the two is true?

Answer: the proposition is true; the converse, "If an animal has four legs, then it is a cow", is false. A dog has four legs but is not a cow, so it is a counterexample to the converse. (Converse and counterexample, page 1.)

Question 4. Proposition: "If two angles are vertically opposite angles, then they are equal." Write its converse and decide whether the converse is true.

Answer: the converse, "If two angles are equal, then they are vertically opposite angles", is false. Counterexample: in a triangle with two equal sides, the angles opposite the equal sides are equal (page 1), but they are at two different corners of the triangle, not vertically opposite angles formed by two crossing lines. (Converse and counterexample, page 1.)

When both are true, and when both are false

Question 5. Here n is a whole number. Proposition P: "If n is even, then n × n is even." Converse Q: "If n × n is even, then n is even." Are both true? What does that tell you about the two statements?

Answer: both are true, so the two statements imply each other. P: an even number times an even number is even, so if n is even, n × n is even. Q: suppose n were odd; then n × n would be odd × odd, which is odd. So if n × n is even, n cannot be odd: it must be even. Since P and its converse are both true, "n is even" and "n × n is even" imply each other. (A proposition and its converse both true, page 3.)

Question 6. Show that the claim "Every number that ends in 1 is prime" is false.

Answer: 551 is a counterexample. It ends in 1, but 551 = 19 × 29, so it has factors other than 1 and itself and is not prime. One counterexample is enough to show the claim is false. (Counterexamples, pages 1 and 2.)

Question 7. Rewrite "A shape has four sides when it is a square" as an if-then proposition. Is it true? Is its converse true?

Answer: "If a shape is a square, then it has four sides": true. Converse, "If a shape has four sides, then it is a square": false. "Y when X" means "if X then Y", so here X is "it is a square" and Y is "it has four sides". Every square has four sides, but a trapezium with only one pair of parallel sides also has four sides and is not a square, so it is a counterexample to the converse. ("Y when X", page 5; converse and counterexample, page 1.)

Question 8. Proposition P: "If two rectangles have the same perimeter, then they have the same area." Converse Q: "If two rectangles have the same area, then they have the same perimeter." Use these rectangles to decide whether each is true: a 26 cm × 36 cm rectangle and a 31 cm square; a 20 cm × 45 cm rectangle and a 30 cm square.

Answer: both P and Q are false. P: the 26 × 36 rectangle has perimeter 26 + 36 + 26 + 36 = 124 cm and the 31 cm square has perimeter 31 + 31 + 31 + 31 = 124 cm, the same, but their areas are 26 × 36 = 936 cm² and 31 × 31 = 961 cm², different. Q: the 20 × 45 rectangle and the 30 cm square both have area 900 cm² (20 × 45 = 30 × 30 = 900), but their perimeters are 20 + 45 + 20 + 45 = 130 cm and 30 + 30 + 30 + 30 = 120 cm, different. So this proposition and its converse are both false. (A proposition and its converse can both be false, page 3; chapter summary, page 7.)

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Sources

All opened and read on 22 October 2026. Page numbers are the printed page numbers of the textbook; Part II starts again at page 1, so printed page = PDF page + 0.

The definitions are from the textbook pages above. The questions, worked answers and checks are our own, and every number was re-computed by script. This guide is general information; follow your teacher and the textbook.