Class 9 Mathematics NCERT Ganita Manjari Chapter 9
Propositions and their Converses

1. If two angles of a triangle are equal, are the opposite sides equal?

Answer: Yes.

Let ∠B = ∠C in △ABC. Draw AD perpendicular to BC. Then the two right triangles formed have one equal acute angle and the common side AD. Hence they are congruent by ASA, giving AB = AC.

2. Can both a proposition and its converse be false?

Answer: Yes.

Consider: “If a positive integer is even, then it is a multiple of 3.” This is false because 2 is even but is not a multiple of 3.

Its converse, “If a positive integer is a multiple of 3, then it is even,” is also false because 3 is a multiple of 3 but is not even.

3. Give examples where a proposition is true but its converse is false.

Geometry: If a quadrilateral is a square, then all its angles are equal. The converse is false because a non-square rectangle also has four equal angles.

Number theory: If a number is a multiple of 6, then it is a multiple of 3. The converse is false; 9 is a multiple of 3 but not of 6.

Real life: If it rains on an uncovered road, the road becomes wet. A wet road does not necessarily mean that it has rained; it may have been washed by a tanker.

Question 1

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

Proposition: True.

When a transversal cuts two parallel lines, corresponding angles are equal.

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

Converse: True.

Equality of a pair of corresponding angles is a criterion for two lines to be parallel.

Question 2

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

Proposition: True.

Each angle of a square is 90°.

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

Converse: False.

Question 3

Given △ABC, the angle bisectors at B and C meet at the incentre I and are extended to meet the opposite sides at E and F.

Proposition: If AB = AC, then IE = IF.

Proposition: True.

Since AB = AC, the base angles are equal: ∠ABC = ∠BCA. Therefore their bisected angles are equal. In △BIC, this gives BI = CI. Now compare △CIE and △BIF. They have two equal angles and the corresponding side CI = BI. Hence the triangles are congruent by ASA, so IE = IF.

Converse: If IE = IF, then AB = AC.

Converse: False.

Question 4

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

Proposition: True.

Adding the same number a to equal numbers preserves equality.

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

Converse: True.

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

Question 5

Proposition: If a and b are perfect squares, then ab is a perfect square.

Proposition: True.

Let a = m² and b = n². Then ab = m²n² = (mn)², which is a perfect square.

Converse: If ab is a perfect square, then a and b are perfect squares.

Converse: False.

Question 6

Here x and y are real numbers.

Proposition: If x = y, then x² = y².

Proposition: True.

Equal real numbers have equal squares.

Converse: If x² = y², then x = y.

Converse: False.

Question 7

Proposition: If x = y, then x³ = y³.

Proposition: True.

Equal real numbers have equal cubes.

Converse: If x³ = y³, then x = y.

Converse: True.

The cube function is one-to-one over the real numbers. Taking the real cube root of both sides gives x = y.

Question 8

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

Proposition: True.

If n = 24k, then n = 4(6k) and n = 6(4k). Hence n is divisible by both 4 and 6.

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

Converse: False.

Question 9

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

Proposition: True.

If n = 60k, then n = 5(12k) and n = 12(5k).

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

Converse: True.

Since 5 and 12 are coprime, LCM(5, 12) = 60. Therefore every number divisible by both 5 and 12 is divisible by 60.

Question 10

Proposition: If n is the square of a prime number, then it has exactly 3 factors.

Proposition: True.

If n = p², where p is prime, its positive factors are exactly 1, p and p².

Converse: If n has exactly 3 factors, then it is the square of a prime number.

Converse: True.

For a positive integer with exactly three positive factors, the factors must be 1, p and n. Since p divides n and there is no other factor between 1 and n, n ÷ p = p. Hence n = p². Moreover, p must be prime; otherwise p would have an additional factor and n would have more than three factors.

Question 11

Proposition: If n is a product of two unequal prime numbers, then it has exactly 4 divisors.

Proposition: True.

Let n = pq, where p and q are distinct primes. Its positive divisors are 1, p, q and pq.

Converse: If n has exactly 4 divisors, then it is a product of two unequal prime numbers.

Converse: False.

Question 12

Proposition: If n and n + 3 have no factors in common, then n is not a multiple of 3.

Proposition: True.

If n were a multiple of 3, then n + 3 would also be a multiple of 3. Thus 3 would be a common factor, contradicting the given condition.

Converse: If n is not a multiple of 3, then n and n + 3 have no common factor other than 1.

Converse: True.

If d is a common factor of n and n + 3, then d also divides their difference:

(n + 3) − n = 3.

Hence d can only be 1 or 3. Since n is not divisible by 3, d cannot be 3. Therefore the only common factor is 1.

Question 13

Find counterexamples to the following claims.

(i) All numbers of the form 4n² + 1 are prime.

(ii) All numbers of the form n² + n + 11 are prime.

(iii) All numbers of the form 4ⁿ + 3 are prime.

Question 14

Find counterexamples to the following statements.

(i) If n is a prime number, then 2ⁿ − 1 is a prime number.

(ii) If n is an even number, then 2ⁿ + 1 is a prime number.

Question 15

Consider: “If a number is divisible by 8, then it is divisible by both 2 and 4.”

(i) Justify the statement.

Let n = 8k. Then n = 2(4k) and n = 4(2k). Therefore n is divisible by both 2 and 4. The statement is true.

(ii) Is it enough to check divisibility by 2 and 4 to conclude that a number is divisible by 8?

No.

The standard divisibility test for 8 is to check whether the number formed by its last three digits is divisible by 8.

Question 16

Express the relationship between divisibility by 3 and the sum of the digits using “If-then” sentences.

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

Statement 2: If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3.

Both statements are true.

They are converses of each other. For example, 123 has digit sum 1 + 2 + 3 = 6, which is a multiple of 3, and 123 = 3 × 41.

Question 17

Suppose Q is a category of quadrilaterals and two sticks are used as diagonals.

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

(a) The two sticks should be of equal length, because they represent the two diagonals and the diagonals must be equal.

(b) The way in which the sticks are put together may matter. Equal diagonals alone do not necessarily guarantee that the resulting quadrilateral belongs to Q. For example, rectangles have equal diagonals, but their diagonals must also bisect each other.

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

(a) The two sticks should be equal in length so that the constructed quadrilateral has equal diagonals.

(b) With equal sticks, any arrangement that forms a proper quadrilateral satisfies the stated condition, because its diagonals are equal. No additional condition about their angle or intersection point follows from the given implication.