Exercise Set 9.1 Class 9 Maths Ganita Manjari Part 2

 

Exercise Set 9.1 — Question 1

Question 1

Proposition: If two lines are parallel, then the corresponding angles formed by a transversal are equal.

Frame the converse and determine whether the proposition and its converse are true or false. Justify your answer.

Solution

1. Proposition

The given proposition is:

If two lines are parallel, then their corresponding angles formed by a transversal are equal.

Let two parallel lines \(l\) and \(m\) be cut by a transversal \(t\).

Since \[ l \parallel m, \] the corresponding angles are equal by the Corresponding Angles Theorem.

For example, \[ \angle 1 = \angle 2 \] where \(\angle 1\) and \(\angle 2\) are corresponding angles.

Therefore, the proposition is TRUE. ✓

2. Converse

To form the converse, interchange the condition and the conclusion.

Converse: If the corresponding angles formed by a transversal are equal, then the two lines are parallel.

Suppose the corresponding angles are equal:

\[ \angle 1 = \angle 2 \]

By the converse of the Corresponding Angles Theorem, if a transversal makes equal corresponding angles with two lines, then the two lines are parallel.

\[ \therefore\quad l \parallel m \]

Therefore, the converse is TRUE. ✓

Final Answer

Statement	Result
Proposition	TRUE ✓
Converse	TRUE ✓

Conclusion: Both the proposition and its converse are true.

Exercise Set 9.1 — Question 2

Question 2

Proposition: If a quadrilateral is a square, then all its angles are equal.

Frame the converse and determine whether the proposition and its converse are true or false. Justify your answer and give a counterexample where necessary.

Solution

1. Proposition

The given proposition is:

If a quadrilateral is a square, then all its angles are equal.

Let \(ABCD\) be a square. 

By the definition of a square, all four angles are right angles. Therefore,

\[ \angle A=\angle B=\angle C=\angle D=90^\circ \]

Therefore, the proposition is TRUE. ✓

2. Converse

To form the converse, interchange the condition and the conclusion.

Converse: If all the angles of a quadrilateral are equal, then the quadrilateral is a square.

Let all four angles of a quadrilateral be equal to \(x\).

The sum of the interior angles of a quadrilateral is \[ 360^\circ. \]

Therefore,

\[ x+x+x+x=360^\circ \] \[ 4x=360^\circ \] \[ x=90^\circ \]

Thus, a quadrilateral whose four angles are equal has four right angles.

However, this does not mean that all four sides are equal. Therefore, it need not be a square.

3. Counterexample

Consider a rectangle of dimensions 6 cm × 4 cm. 

All its angles are right angles:

\[ 90^\circ,\quad90^\circ,\quad90^\circ,\quad90^\circ \]

Therefore, all its angles are equal.

But its sides are not all equal:

\[ 6\text{ cm}\ne4\text{ cm} \]

Hence, this quadrilateral is a rectangle, but it is not a square.

Therefore, the converse is FALSE. ✗

Counterexample: A non-square rectangle, such as a \(6\text{ cm}\times4\text{ cm}\) rectangle.

Final Answer

Statement	Result
Proposition	TRUE ✓
Converse	FALSE ✗

Conclusion: The proposition is true, but its converse is false because a rectangle can have four equal angles without being a square.