CONDITIONAL AND BICONDITIONAL STATEMENTS WORKSHEET

Problem 1 :

Rewrite the following conditional statements in if-then form.

(i) Two points are collinear if they lie on the same line.

(ii) A number is divisible by 9 is also divisible by 3.

(iii) All sharks have a boneless skeleton.

Problem 2 :

Write a counter example to show that the following conditional statement is false.

If x² = 25, then x = 5

Problem 3 :

Write the converse of the following conditional statement.

Statement :

"If two segments have the same length, then they are congruent"

Problem 4 :

Write (a) inverse, (b) converse, (c) contrapositive of the following statement.

Statement :

"If there is snow on the ground, the flowers are not in bloom"

Problem 5 :

Whether each statement about the diagram is true. Explain your answer using the definitions you have learned.

(i)  Points D, X and B are collinear.

(ii)  AC is perpendicular to DB.

(iii)  ∠AXB is adjacent to ∠CXD.

Problem 6 :

Write the following biconditional statement as a conditional statement and its converse.

"Three lines are coplanar if and only if they lie in the same plane"

Problem 7 :

Consider the following statement :

x = 3 if and only if x² = 9

(i) Is this a biconditional statement?

(ii) Is the statement true?

Problem 8 :

Each of the following statements is true. Write the converse of each statement and decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample.

(i) If two points lie in a plane, then the line containing them lies in the plane.

(ii) If a number ends in 0, then the number is divisible by 5.

1. Answer :

(i) If two points lie on the same line, then they are collinear.

(ii) If a number is divisible by 9, then it is divisible by 3.

(iii) If a fish is a shark, then it would have a boneless skeleton.

2. Answer :

As a counter example, let us take x = -5.

The hypothesis is true, because (-5)² = 25. But, the conclusion is false, because it is given x = 5.

It implies that the given conditional statement is false.

3. Answer :

Converse :

"If two segments are congruent, then they have the same length"

4. Answer :

(a) Inverse :

"If there is no snow on the ground, the flowers are in bloom"

(b) Converse :

"If flowers are not in bloom, then there is snow on the ground"

(b) Contrapositive :

"If flowers are in bloom, then there is no snow on the ground"

5. Answer :

(i) This statement is true. Two or more points are collinear, if they lie on the same line. The points D, X and B all lie on line DB. So they are collinear.

(ii) This statement is true. The right angle symbol in the diagram indicates that the lines AC and BD intersect to form a right angle. So, the lines are perpendicular.

(iii) This statement is false. By definition, adjacent angles must share a common side. Because ∠AXB and ∠CXD do not share a common side, they are adjacent.

6. Answer :

Conditional Statement :

If three lines are coplanar, then they lie in the same plane.

Converse :

If three lines lie in the same plane, then they are coplanar.

7. Answer :

(i) The statement is biconditional because it contains “if and only if.”

(ii) The statement can be rewritten as the following statement and its converse.

Conditional statement :

If x = 3, then x² = 9

Converse :

If x² = 9, then x = 3

The first of these statements is true, but the second is false. Because, if x² = 9, then x = 3 or -3.

So, the biconditional statement is false.

8. Answer :

Solution (i) :

Converse :

(i) If a line containing two points lies in a plane, then the points lie in the plane.

The converse is true, as shown in the diagram. So, it can be combined with the original statement to form the true biconditional statement written below.

Biconditional statement :

Two points lie in a plane, if and only if the line containing them lies in the plane.

Solution (ii) :

Converse :

If a number is divisible by 5, then the number ends in 0. The  converse is false. As a counterexample, consider the number 15. It is divisible by 5, but it does not end in 0, as shown below.

20 ÷ 5  =  4

25 ÷ 5  =  5

30 ÷ 5  =  6

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