CLASS 9 · GANITA MANJARI PART II · CHAPTER 9

Propositions and their Converses

Learn when a statement and its converse are true. Short proofs and counterexamples come first; use the explanation button to follow the reasoning.

32 worked answersAll 1 exercise sets

Read the NCERT questions ↗← All chapters

32 worked answers · short steps first, explanations when you need them

Exercise Set 9.1

Textbook pages 5–7

Question 1

State the converse of the corresponding-angle property for parallel lines and decide whether both statements hold.

Steps only

Parallel lines ⇒ equal corresponding angles.

Equal corresponding angles ⇒ parallel lines.

Answer

Both statements are true.

Question 2

A square has four equal angles. Is the reverse statement true for a quadrilateral?

Steps only

A square has four right angles.

360° ÷ 4 = 90°

A 6 cm by 3 cm rectangle has equal angles but is not a square.

Answer

Forward: true. Converse: false.

★ Question 3

With the angle bisectors meeting at I and continuing to E and F, examine AB = AC ⇒ IE = IF and its converse.

Steps only

AB = AC: reflection in AI exchanges E and F, so IE = IF.

Counterexample: A(0,0), B(2,0), C(1/2, √3/2).

AB = 2; AC = 1

IE² = IF² = 14 − 8√3

A scalene triangle with equal angle-bisector tails · Original Learnivelle diagram. Equal scale on both axes.

A(0, 0) · B(2, 0) · C(1/2, 0.866 (approx.)) · I(0.634 (approx.), 0.366 (approx.)) · E(0.2679 (approx.), 0.4641 (approx.)) · F(0.7321 (approx.), 0)

Answer

Forward: true. Converse: false.

Question 4

For real numbers a, x and y, test x = y ⇒ a + x = a + y and its converse.

Steps only

x = y ⇒ a + x = a + y

a + x = a + y ⇒ x = y

Answer

Both statements are true.

Question 5

Test whether two perfect squares have a square product, and whether a square product forces both factors to be squares.

Steps only

a = r²; b = s² ⇒ ab = (rs)²

2 × 8 = 16 = 4²

Neither 2 nor 8 is a perfect square.

Answer

Forward: true. Converse: false.

Question 6

For real x and y, test x = y ⇒ x² = y² and the converse.

Steps only

x = y ⇒ x² = y²

1² = (−1)² = 1

1 ≠ −1

Answer

Forward: true. Converse: false.

Question 7

For real x and y, test x = y ⇒ x³ = y³ and the converse.

Steps only

x = y ⇒ x³ = y³

x³ − y³ = (x − y)(x² + xy + y²) = 0

x² + xy + y² = (x + y/2)² + 3y²/4

The second factor is positive unless x = y = 0.

Answer

Both statements are true.

Question 8

Test the statement that a multiple of 24 is a multiple of both 4 and 6, and test its converse.

Steps only

24 is divisible by 4 and 6.

LCM(4, 6) = 12

12 is divisible by 4 and 6, but not by 24.

Answer

Forward: true. Converse: false.

Question 9

Test the statement that a multiple of 60 is a multiple of both 5 and 12, and test its converse.

Steps only

60 is divisible by 5 and 12.

LCM(5, 12) = 60

Answer

Forward: true. Converse: true.

Question 10

Test whether a prime square has exactly three positive divisors, and whether every number with three divisors is a prime square.

Steps only

p prime ⇒ divisors of p²: 1, p, p²

Three divisors leave one nontrivial divisor p, paired with itself.

n = p²; p must be prime

Answer

Both statements are true.

Question 11

Test whether a product of two different primes has four divisors, and test the converse.

Steps only

p ≠ q, both prime: divisors of pq are 1, p, q, pq

8: divisors are 1, 2, 4, 8

8 = 2³

Answer

Forward: true. Converse: false.

Question 12

For positive integer n, test whether coprime n and n + 3 imply that n is not divisible by 3, and test the converse.

Steps only

HCF(n, n + 3) = HCF(n, 3)

Since 3 is prime, the HCF is 1 exactly when 3 does not divide n.

Answer

Both statements are true.

Question 13(i)

Give a counterexample to the claim that 4n² + 1 is always prime.

Steps only

n = 4

4n² + 1 = 65 = 5 × 13

Answer

n = 4 gives the composite number 65.

Question 13(ii)

Give a counterexample to the claim that n² + n + 11 is always prime.

Steps only

n = 10

n² + n + 11 = 121 = 11 × 11

Answer

n = 10 gives the composite number 121.

Question 13(iii)

Give a counterexample to the claim that 4n + 3 is always prime.

Steps only

n = 3

4n + 3 = 15 = 3 × 5

Answer

n = 3 gives the composite number 15.

Question 14(i)

Give a counterexample to the claim that 2ⁿ − 1 is prime when n is prime.

Steps only

n = 11

2ⁿ − 1 = 2047 = 23 × 89

Answer

n = 11 gives the composite number 2047.

Question 14(ii)

Give a counterexample to the claim that 2ⁿ + 1 is prime when n is even.

Steps only

n = 6

2ⁿ + 1 = 65 = 5 × 13

Answer

n = 6 gives the composite number 65.

Question 15(i)

Explain why every multiple of 8 is divisible by both 2 and 4.

Steps only

n = 8k = 2(4k) = 4(2k)

Answer

The statement is true.

Question 15(ii)

Does checking divisibility by both 2 and 4 guarantee divisibility by 8?

Steps only

4 ÷ 2 = 2; 4 ÷ 4 = 1

4 ÷ 8 = 1/2

Answer

No; 4 is a counterexample.

Question 16

Express both directions of the divisibility-by-3 digit-sum test as if–then statements.

Steps only

If a number is divisible by 3, its digit sum is divisible by 3.

If its digit sum is divisible by 3, the number is divisible by 3.

Answer

Both implications are true.

Question 17(i)(a)

If every quadrilateral of type Q has equal diagonals, must sticks used as its diagonals be equal?

Steps only

Q ⇒ equal diagonals: equality is necessary.

Answer

Yes. Equal sticks are required.

Question 17(i)(b)

If Q implies equal diagonals, does any placement of equal sticks guarantee a quadrilateral of type Q?

Steps only

Equal diagonals alone do not imply Q.

For example, an isosceles trapezium can have equal diagonals without being a rectangle.

Answer

Placement may matter; the given statement gives no guarantee.

Question 17(ii)(a)

If equal diagonals imply type Q, how should you choose the two diagonal sticks?

Steps only

Choose equal sticks to ensure equal diagonals and therefore Q.

Answer

Equal sticks guarantee Q; the statement does not rule out other quadrilaterals of type Q.

Question 17(ii)(b)

If equal diagonals imply Q, does the angle or crossing position of equal diagonal sticks impose another condition?

Steps only

For any placement making a valid quadrilateral, the equal diagonals imply Q.

Answer

No extra restriction follows, apart from forming a valid quadrilateral.

Think and Reflect

Textbook pages 1–5

Page 1

Prove that equal angles at B and C in triangle ABC imply AB = AC.

Steps only

Draw the angle bisector AD from A to BC.

∠BAD = ∠CAD; ∠ABD = ∠ACD; AD = AD

△ABD ≅ △ACD (AAS)

AB = AC

Answer

The converse is true: AB = AC.

Page 2 · Multiples

Explain 6 divides n ⇒ 3 divides n, and decide whether the reverse is true.

Steps only

n = 6k = 3(2k)

3 is divisible by 3 but not by 6.

Answer

Forward true; converse false.

Page 2–3 · Factors

Explain both directions linking perfect squares with an odd number of positive divisors.

Steps only

Divisors pair as d and n/d.

A square has exactly one repeated pair (√n, √n).

Answer

Both directions are true.

Page 3 · Areas

Test equal triangle areas ⇒ congruent triangles, and the converse.

Steps only

(1/2) × 4 × 3 = 6

(1/2) × 6 × 2 = 6

Right triangles with legs (4,3) and (6,2) have unequal side lengths.

Answer

Equal area does not force congruence; congruence does force equal area.

Page 3 · Both false

Give a proposition whose converse is also false.

Steps only

Claim: every even positive integer is a multiple of 3. Counterexample: 2.

Converse: every multiple of 3 is even. Counterexample: 3.

Answer

Both statements are false.

Page 4

Prove that a triangle with side lengths a, b, c satisfying a² + b² = c² is right-angled.

Steps only

Construct a right triangle with perpendicular sides a and b.

Hypotenuse = √(a² + b²) = c

The two triangles are congruent by SSS.

Answer

The angle opposite side c is 90°.

Page 5 · 1

Give an everyday implication that holds but whose converse need not hold.

Steps only

In a class marked present only after arrival: attending implies being registered.

A registered pupil may be absent.

Answer

Registered does not necessarily mean attending.

Page 5 · 2

Give a number example and a geometry example with a true forward implication and a false converse.

Steps only

Multiple of 10 ⇒ even; 2 disproves the converse.

Square ⇒ rectangle; a 6 by 3 rectangle disproves the converse.

Answer

Both examples have a true statement and a false converse.