GANITA MANJARI · GRADE 9 · PART II · CHAPTER 9
Think first. Use AI purposefully. Verify always.
TEXTBOOK CONNECTION
Ganita Manjari · Grade 9 · Part II · Chapter 9. Use your textbook for the complete questions. These are topic references and original AIMS guidance, not reproduced exercises.
Textbook Guided Example
Source page 1 · REVERSE
Everyday cause and a possible alternative cause
Identify the stated direction before reversing it. A different cause can defeat the converse.
Original AIMS parallel: a club member gets a card, but a visitor may receive the same card.
Textbook Guided Example
Source page 2 · REVERSE
Divisibility in two directions
Write a general multiple as a product, then look for a multiple of the smaller divisor that fails the reverse.
Keep the original textbook numbers in your book; AIMS uses fresh numerical examples in REVERSE.
Textbook Guided Example
Source page 2 · REASON
Perfect squares and factor partners
Give distinct reasons for the odd-factor-count direction and the square direction.
The positive self-partner is unique; include n = 1.
Textbook Guided Example
Source page 3 · APPLY
Equal triangle areas and congruence
Equal area need not determine shape and size. Congruence does preserve area.
AIMS triangles with perpendicular legs 3,8 and 4,6 each have area 12 but differ in side lengths.
Textbook Guided Example
Source page 4 · PROVE
The converse theorem and SSS
Construct a right triangle on the two shorter side lengths, then match all three sides.
Only the constructed triangle starts with a known right angle.
Exercise Practice
Source page 5 · APPLY
Parallel lines and corresponding angles
Keep the same pair of lines and transversal in both directions.
The corresponding-angle theorem and its converse justify the two directions.
Exercise Practice
Source page 5 · COMPARE
Squares and equal-angle quadrilaterals
Check whether angle information also forces all four side lengths equal.
A non-square rectangle distinguishes the two directions.
Exercise Practice
Source page 6 · APPLY
Equality after adding the same quantity
Use addition forward and subtraction in the reverse direction.
The operation uses the same added quantity on both sides.
Exercise Practice
Source page 6 · APPLY
Products of perfect squares
Writing the factors as squares proves one direction. Test non-square factors for the reverse.
Original AIMS example: 3 × 12 = 36, although neither factor is square.
Exercise Practice
Source page 6 · APPLY
Equality of real squares
Do not drop the possibility of opposite real inputs.
Equal real squares need not mean equal inputs.
Exercise Practice
Source page 6 · APPLY
Equality of real cubes
Cubing preserves order on the real number line.
Unequal real inputs cannot have equal cubes.
Exercise Practice
Source page 6 · TEST
Divisibility by 24 versus divisibility by 4 and 6
A shared multiple of 4 and 6 need not contain every factor required by 24.
Test the reverse separately rather than multiplying the two divisors blindly.
Exercise Practice
Source page 6 · APPLY
Divisibility by 60 versus divisibility by 5 and 12
The factors 5 and 12 share no prime factor.
Justify why both divisibilities here do supply the full product.
Exercise Practice
Source page 6 · REASON
Prime squares and exactly three factors
List the factors of a prime square, then justify the middle factor for the reverse.
A composite middle factor would introduce more positive divisors.
Exercise Practice
Source page 6 · MASTER
Distinct prime products and exactly four divisors
Test prime cubes as well as products of different primes.
Four divisors do not force two distinct primes.
Exercise Practice
Source page 6 · APPLY
Common factors of n and n + 3
Here “no factors in common” means no common positive factor greater than 1. Any shared divisor divides their difference.
The difference is 3; use this fact for both directions.
Exercise Practice
Source page 6 · TEST
Prime-generating claims with square and exponential expressions
Seek a permitted input giving a composite result and show a nontrivial factorisation.
The third source expression has an exponent on 4; it is not 4 times n.
Exercise Practice
Source page 6 · TEST
Prime claims involving powers of 2
Keep the prime/even input restrictions and check the powers accurately.
Do not confuse an exponent with multiplication; supporting small cases are not a proof.
Exercise Practice
Source page 6 · MASTER
A proposed divisibility shortcut for 8
Separate “multiple of 8” from “multiple of both 2 and 4”.
A condition required in the forward direction may fail as a reverse test.
Exercise Practice
Source page 6 · REVERSE
Digit-sum test for divisibility by 3
Express each direction with an if–then sentence.
Keep the same positive integer in both sentences; the starting condition and conclusion exchange places.
Exercise Practice
Source page 7 · MASTER
Quadrilateral diagonal conditions · Figure 9.2
In the first direction equal sticks are required but do not alone guarantee the category. In the reverse direction equal sticks suffice when a genuine quadrilateral is formed.
The second direction alone does not assert that every quadrilateral of the category has equal diagonals.
