Class 9 Ganita Manjari
Chapter 9: Propositions and their Converses
Part 7
[cite: 6]
Exercise 9.1: The Converses
Now it is time to put everything we've learned into practice! In Exercise 9.1, you are tasked with framing the exact converse for various geometrical and algebraic propositions. [cite: 6]
More importantly, you must determine whether the original statement and its new converse are true or false. [cite: 6] Remember, if you claim a statement is false, you must provide a solid counterexample to prove it. In algebra, negative numbers are often the secret weapons that break false converses! Let's dive in.
Now it is time to put everything we've learned into practice! In Exercise 9.1, you are tasked with framing the exact converse for various geometrical and algebraic propositions. [cite: 6]
More importantly, you must determine whether the original statement and its new converse are true or false. [cite: 6] Remember, if you claim a statement is false, you must provide a solid counterexample to prove it. In algebra, negative numbers are often the secret weapons that break false converses! Let's dive in.
1 Exercise 9.1 - Question 2
For the proposition: "If a quadrilateral is a square, then all its angles are equal." [cite: 6]
Frame its converse. Then, determine if both the original proposition and the converse are true or false. Justify your answers and provide a counterexample if false. [cite: 6]
Frame its converse. Then, determine if both the original proposition and the converse are true or false. Justify your answers and provide a counterexample if false. [cite: 6]
Concept:
1
A square is a highly specific shape: it has four equal sides AND four equal angles. Therefore, any square will satisfy the condition of having equal angles. [cite: 6]
2
However, just having equal angles does not force a shape to have equal sides! [cite: 6]
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Hint:
1
To write the converse, swap the "If" and "Then". [cite: 6]
2
Converse: "If all angles of a quadrilateral are equal, then it is a square." [cite: 6]
3
Look at the Notebook section above! What other shape has four $90^\circ$ angles? [cite: 6]
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Mistakes to Avoid:
!
Assuming all rectangles are squares: All squares are rectangles, but not all rectangles are squares! A rectangle acts as the perfect counterexample because it fulfills the "equal angle" requirement but fails the "square" requirement.
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Solution:
1. Original Proposition: "If a quadrilateral is a square, then all its angles are equal." [cite: 6]
Truth: True. Justification: By geometric definition, a square is a regular quadrilateral with four equal sides and four equal angles (\(90^\circ\) each). [cite: 6]
2. The Converse: "If all angles of a quadrilateral are equal, then it is a square." [cite: 6]
Truth: False. [cite: 6]
Counterexample: A rectangle is a quadrilateral where all four angles are equal (\(90^\circ\)), but its adjacent sides are not necessarily equal in length, meaning it is not a square. [cite: 6]
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2 Exercise 9.1 - Question 6
For the proposition: "If \(x = y\), then \(x^2 = y^2\) (where \(x\) and \(y\) are real numbers)." [cite: 6]
Frame its converse. Then, determine if both the original proposition and the converse are true or false. Justify your answers and provide a counterexample if false. [cite: 6]
Frame its converse. Then, determine if both the original proposition and the converse are true or false. Justify your answers and provide a counterexample if false. [cite: 6]
Concept:
1
Squaring an equation on both sides is a valid algebraic operation. If two things are identical, their squares will be identical. [cite: 6]
2
However, taking the square root introduces a $\pm$ sign! Both positive and negative numbers yield positive squares. [cite: 6]
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Hint:
1
Converse: "If \(x^2 = y^2\), then \(x = y\)." [cite: 6]
2
Can you find two different numbers that equal the same thing when squared? Try 3 and -3! [cite: 6]
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Mistakes to Avoid:
!
Forgetting negative numbers: If you only test positive numbers (e.g., $4^2 = 4^2$), the converse looks true! You must test all "real numbers," which includes negatives, to find the counterexample. [cite: 6]
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Solution:
1. Original Proposition: "If \(x = y\), then \(x^2 = y^2\)." [cite: 6]
Truth: True. Justification: If two variables hold the exact same numerical value, multiplying that value by itself will yield the exact same result on both sides. [cite: 6]
2. The Converse: "If \(x^2 = y^2\), then \(x = y\)." [cite: 6]
Truth: False. [cite: 6]
Counterexample: Let \(x = 2\) and \(y = -2\). [cite: 6]
Here, \(2^2 = 4\) and \((-2)^2 = 4\). Therefore, \(x^2 = y^2\) is satisfied.
However, \(2 \neq -2\), so \(x\) does not equal \(y\). This disproves the converse! [cite: 6]
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Objective 1: Multiplying Perfect Squares (Q5)
Consider the converse of Question 5: "If the product \(ab\) is a perfect square, then both \(a\) and \(b\) must be perfect squares." [cite: 6]
Which of the following acts as a valid counterexample proving this converse FALSE?
Which of the following acts as a valid counterexample proving this converse FALSE?
Concept:
1
If $a$ and $b$ are squares, their product is always a square. [cite: 6] But the converse asks: if a product is a square, do its parts have to be squares?
2
To break the converse, find two non-square numbers that multiply together to make a perfect square. [cite: 6]
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Hint:
1
Look for an option where \(a \times b\) equals 16, 25, or 36. [cite: 6]
2
Check option C: \(2 \times 8 = 16\). Are 2 and 8 perfect squares? [cite: 6]
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Mistakes to Avoid:
!
Picking Option A: Option A (4 and 9) provides an example of the proposition working correctly. It does not provide a counterexample that breaks the converse! [cite: 6]
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Solution:
The converse statement is "If \(ab\) is a perfect square, then \(a\) and \(b\) are perfect squares." [cite: 6]
We need a scenario where \(ab\) is a square, but \(a\) and \(b\) are not.
In Option C, \(a = 2\) and \(b = 8\).
The product \(ab = 16\), which is a perfect square (\(4^2\)).
However, neither 2 nor 8 are perfect squares themselves. This perfectly breaks the rule! [cite: 6]
Correct Option: (C)
Objective 2: The Cubed Equation (Q7)
For the proposition: "If \(x = y\), then \(x^3 = y^3\)". [cite: 6] Is the converse of this specific statement true or false for real numbers?
Concept:
1
While $x^2 = y^2$ fails its converse due to positive/negative overlaps (e.g. $2^2 = (-2)^2 = 4$), cubes do not have this problem! [cite: 6]
2
A negative number cubed stays negative (e.g., $(-2)^3 = -8$). A positive number cubed stays positive (e.g., $2^3 = 8$). Therefore, every real cube has exactly one unique real cube root. [cite: 6]
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Hint:
1
The converse is: "If \(x^3 = y^3\), then \(x = y\)." [cite: 6]
2
If \(x^3 = -27\) and \(y^3 = -27\), then both \(x\) and \(y\) MUST be -3. There is no other option! [cite: 6]
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Mistakes to Avoid:
!
Assuming all exponents behave like squares: Just because the converse failed for squares (Question 6) does not mean it fails for cubes (Question 7). Odd powers preserve the sign, making the converse true! [cite: 6]
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Solution:
The converse is "If \(x^3 = y^3\), then \(x = y\)." [cite: 6]
Unlike squaring, which destroys the negative sign (\(-2 \times -2 = +4\)), cubing a number preserves its original sign (\(-2 \times -2 \times -2 = -8\)). [cite: 6]
Because every real number has one and only one unique real cube root, if \(x^3\) and \(y^3\) yield the exact same value (including the sign), then \(x\) and \(y\) must be the exact same number.
Therefore, the converse is True. [cite: 6]
Correct Option: (B)
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Exercise 9.1 - Question 1
For the proposition: "If two lines are parallel, then the corresponding angles formed by a transversal are equal." [cite: 6]
Frame its converse. Then, determine if both the original proposition and the converse are true or false. Justify your answers. [cite: 6]
Frame its converse. Then, determine if both the original proposition and the converse are true or false. Justify your answers. [cite: 6]
Concept:
1
This is a fundamental theorem from geometry regarding transversals cutting across lines. [cite: 6]
2
In Euclidean geometry, the corresponding angles axiom works perfectly in both directions! [cite: 6]
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Hint:
1
Swap the parts to build the converse: "If the corresponding angles formed by a transversal are equal, then..." [cite: 6]
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Mistakes to Avoid:
!
Assuming all geometry converses are false: We saw in Question 2 (Squares/Rectangles) that a geometric converse was false. But this axiom about parallel lines is universally true in both directions. Don't jump to conclusions without testing! [cite: 6]
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Solution:
1. Original Proposition: "If two lines are parallel, then the corresponding angles formed by a transversal are equal." [cite: 6]
Truth: True. Justification: This is the foundational Corresponding Angles Axiom in geometry. [cite: 6]
2. The Converse: "If the corresponding angles formed by a transversal are equal, then the two lines are parallel." [cite: 6]
Truth: True. [cite: 6]
Justification: This is the Converse of the Corresponding Angles Axiom, which is a mathematically proven theorem used constantly to prove that lines are parallel. [cite: 6]
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