Get the simplified Extra Questions for Class 9 Maths and Ganita Manjari Class 9 Maths Chapter 9 Propositions and their Converses Extra Questions with complete explanation.
Class 9 Propositions and their Converses Extra Questions
Extra Questions on Propositions and their Converses Class 9
Class 9 Ganita Manjari Chapter 9 Extra Questions
Question 1.
Write the converse of the following proposition:
“If a triangle is equilateral, then all its sides are equal.”
Solution:
Converse: If all sides of a triangle are equal, then the triangle is equilateral.
Question 2.
Determine whether the following proposition is true or false:
“If a number is divisible by 45, then it is divisible by 15.”
Solution:
The proposition is true.
Let the number be:
n = 45k
Since, 45k = 15 × 3k
Therefore, n is divisible by 15.
Hence, the proposition is true.
Question 3.
Find a counterexample for:
“If a number is even, then it is divisible by 4.”
Solution:
Take the number 6.
6 is even, but it is not divisible by 4.
Therefore, 6 is a counterexample of the statement, and the given proposition is false.
Propositions and their Converses Class 9 Short Question Answer
Question 1.
Check whether the proposition and converse are true or false:
Proposition: If two angles are supplementary, then their sum is 180°.
Converse: If the sum of two angles is 180°, then the angles are supplementary.
Solution:
Proposition: If two angles are supplementary, then their sum is 180°. ‘
Status: True
Justification: Two angles are called supplementary angles when the sum of their measures is 180°. Therefore, if two angles are supplementary, their sum is 180°.
Converse: If the sum of two angles is 180°, then the angles are supplementary.
Status: True
Justification: If the sum of two angles is 180°, then they satisfy the definition of supplementary angles.
Therefore, both the proposition and its converse are true.
Question 2.
Examine the proposition: “If a quadrilateral is a rectangle, then it is a rhombus.”
Write its converse and determine the truth values.
Solution:
Proposition: If a quadrilateral is a rectangle, then it is a rhombus.
This is false because a rectangle with unequal adjacent sides is not a rhombus.
Converse: If a quadrilateral is a rhombus, then it is a rectangle.
This is also false because a rhombus may not have all angles equal to 90°.
Hence, both the proposition and the converse are false.
Question 3.
Examine the proposition: “If a number is the square of an even number, then it is divisible by 4.”
Solution:
Let the even number be: n = 2k, where k is an integer.
The square of the number is:
n2 = (2k)2
n2 = 4k2
Since, 4k2 = 4 × k2
Therefore, the square of an even number is divisible by 4.
Hence, the proposition is true.
Propositions and their Converses Class 9 Long Question Answer
Question 1.
A student claims: “If two statements have the same conclusion, then one statement must be the converse of the other.”
Do you agree with the student? Justify your answer with an example.
Solution:
The student’s claim is false. Two statements can have the same conclusion but still not be converses.
Consider the statements:
Statement P: If a number is divisible by 10, then it ends with 0.
Statement Q: If a number is divisible by 20, then it ends with 0.
Here,
- Conclusion of P = “number ends with 0”
- Conclusion of Q = “number ends with 0”
Both statements have the same conclusion.
However, the converse of P would be: “If a number ends with 0, then it is divisible by 10.”
This is different from Q. Therefore, having the same conclusion does not make two statements converses of each other.
Frame the converse for each of the propositions in questions 2-3. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.
Question 2.
If a number is the square of an odd integer, then it is odd.
Solution:
Proposition: If a number is the square of an odd integer, then it is odd.
Status: True
Justification: Let the odd integer be: n = 2k + 1, where k is an integer.
The square of n is:
(2k + 1)2 = 4k2 + 4k + 1
= 2(2k2 + 2k) + 1
It is of the form 2m + 1, where m = 2k2 + 2k. So, n2 is an odd number.
Therefore, the proposition is true.
Converse: If an integer is odd, then it is the square of an odd integer.
Status: False
Counterexample: Take the number 15. 15 is odd, But there is no integer whose square is 15.
Therefore, 15 is an odd number but not the square of an odd integer.
Hence, the converse is false.
Question 3.
If two angles are vertically opposite angles, then they are equal.
Solution:
Proposition: If two angles are vertically opposite angles, then they are equal.
Status: True
Justification: When two lines intersect, the pair of vertically opposite angles formed are always equal. Therefore, if two angles are vertically opposite, then they are equal. Hence, the proposition is true.
Converse: If two angles are equal, then they are vertically opposite angles.
Status: False
Counterexample: Consider two adjacent angles 60° and 60°.
They are equal, but they are not vertically opposite angles because they do not lie opposite each other when two lines intersect.
Therefore, equal angles need not always be vertically opposite angles.
Hence, the converse is false.
Propositions and their Converses Class 9 Case Based Questions
Question 1.
During a digital awareness activity, students were asked to identify If-Then statements used in everyday situations and analyse them as propositions.
Riya gave the following examples:
Statement P: If a student completes all assignments, then the student can submit the work on time.
Statement Q: If a user enters the correct PIN, then the ATM allows access to the account.
The teacher explained that an ‘If-Then’ statements have two parts:
- Condition (Hypothesis): The part that comes after If.
- Conclusion: The part that comes after Then.
Students were asked to identify the parts of the statements, write their converses, and examine whether the statements remain true after reversing the condition and conclusion.
(i) Identify the condition and conclusion in Statement P.
(ii) Write the converse of Statement P.
(iii) Analyse whether Statement Q and its converse are true or false. Give justification.
OR
Write another real-life ‘If-Then’ statement where the proposition is true but the converse is false. Explain with a counterexample.
Solution:
(i) Statement P: If a student completes all assignments, then the student can submit the work on time. Condition: A student completes all assignments. Conclusion: The student can submit the work on time.
(ii) The converse is formed by interchanging the condition and conclusion.
Original statement: If a student completes all assignments, then the student can submit the work on time.
Converse: If a student can submit the work on time, then the student has completed all assignments.
(iii) Statement Q: If a user enters the correct PIN, then the ATM allows access to the account.
Status: True
Justification: Entering the correct PIN is required for accessing an ATM account.
Therefore, the statement is true.
Converse: If an ATM allows access to the account, then the user has entered the correct PIN.
Status: True
Justification: In normal conditions, ATM access is allowed only after verification of the correct PIN. Therefore, the converse is also true.
OR
Proposition: If a person is in India, then the person is in Asia.
Status: True
Justification: India is located in Asia, so a person in India will always be in Asia.
Converse: If a person is in Asia, then the person is in India. Status: False
Counterexample: A person living in Nepal is in Asia but not in India.
Therefore, the converse is false. (Examples may vary.)
Propositions and their Converses Class 9 Competency Based Questions
Question 1.
A student writes the statement: “If a number is divisible by 5, then it ends with 0 or 5.”
Check whether the statement is true. Also, write its converse.
Solution:
Proposition: If a number is divisible by 5, then it ends with 0 or 5.
This is true because every multiple of 5 has its last digit as 0 or 5.
Converse: If a number ends with 0 or 5, then it is divisible by 5.
This is also true.
Therefore, both proposition and converse are true.
Question 2.
A farmer observes the following situation: “If there is heavy rainfall, then the soil becomes wet.”
A student writes the converse: “If the soil is wet, then there has been heavy rainfall.”
Analyse both statements and determine whether they are true or false. Justify your answer.
Solution:
Original Statement: If there is heavy rainfall, then the soil becomes wet.
Status: True
Justification: Heavy rainfall adds water to the soil, making it wet.
Converse: If the soil is wet, then there has been heavy rainfall.
Status: False
The soil can become wet due to reasons other than heavy rainfall.
Counterexample: A farmer may use irrigation to water the field, making the soil wet even when there is no rainfall. Therefore, the converse is false.
Question 3.
A teacher gives the statement: “If two angles form a linear pair, then their sum is 180°.”
Write the converse and determine whether both statements are true.
Solution:
Proposition: If two angles form a linear pair, then their sum is 180°.
This is true because angles forming a linear pair are supplementary.
Converse: If the sum of two angles is 180°, then they form a linear pair.
This is false because two supplementary angles may not share a common arm or vertex.
Therefore, the proposition is true but the converse is false.
Propositions and their Converses Class 9 Extra Questions for Practice
Multiple Choice Questions
Question 1.
Which of the following statements is a proposition?
(a) Find the value of x.
(b) Close the door.
(c) A triangle has three sides.
(d) Write the converse of the given statement.
Question 2.
The converse of “If a number is divisible by 2, then it is even” is:
(a) If a number is even, then it is divisible by 2.
(b) If a number is not even, then it is not divisible by 2.
(c) If a number is divisible by 2, then it is not even.
(d) If a number is not divisible by 2, then it is even.
Question 3.
A counterexample is used to:
(a) Prove a statement true
(b) Prove a statement false
(c) Change a statement
(d) Write a formula
Question 4.
Which of the following statements is a counterexample to the proposition:
“If a number is divisible by 5, then it is divisible by 10.”
(a) 10
(b) 15
(c) 20
(d) 30
Assertion Reason Questions
Two statements are given, one labelled as Assertion (A) and the other labelled as Reason (R). Select the correct answer from the options (a), (b), (c), and (d) given below.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.
Question 1.
Assertion (A): A false proposition can be proved wrong by giving a suitable counterexample.
Reason (R): A counterexample is an example that does not satisfy the given statement.
Question 2.
Assertion (A): The converse of a proposition is obtained by changing the order of its condition and conclusion.
Reason (R): If a proposition is written as “If X, then Y”, its converse is written as “If Y, then X”.
Short Answer Type Questions
Question 1.
Examine the converse of:”If a number is divisible by 8, then it is even.”
Question 2.
Define a proposition.
Question 3.
Find whether the statement is true or false:
“If two triangles are similar, then they have the same area.”
Question 4.
Examine the proposition and converse:
Proposition: If a number is a multiple of 10, then it ends with zero.
Converse: If a number ends with zero, then it is a multiple of 10.
Question 5.
Write the converse and examine both the converse and the proposition.
Check whether the following proposition and converse are true or false:
Proposition: If a quadrilateral is a parallelogram, then its opposite sides are parallel.
Long Answer Type Questions
Question 1.
Give examples to show the four possible relationships between a proposition and its converse. For each case, write the truth values of the proposition (P) and its converse (Q), and justify your answer.
Consider the following cases:
(a) P is true and Q is false.
(b) P is true and Q is true.
(c) P is false and Q is true.
(d) P is false and Q is false.
For false statements, give suitable counterexamples.
Question 2.
Analyse the following statement and determine whether it is true or false. Write its converse and examine the truth value of both statements. Give proper justification and a counterexample wherever required.
Proposition: If the sum of the interior angles of a polygon is 360°, then the figure is a quadrilateral.