Question

Which of the following statements are true (T) and which are false (F)? Give reasons.

(i) If two lines are intersected by a transversal, then corresponding angles are equal.
(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
(iii) Two lines perpendicular to the same line are perpendicular to each other.
(iv) Two line parallel to the same line are parallel to each other.
(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.

Answer 1

(i)

Statement: If two lines are intersected by a transversal, then corresponding angles are equal.

False

Reason:

The above statement holds good if the lines are parallel only.

(ii)

Statement: If two parallel lines are intersected by a transversal, then alternate interior angles are equal.

True

Reason:

Let l and m are two parallel lines.

And transversal t intersects l and m making two pair of alternate interior angles, ,and,.

We need to prove that and .

We have,

(Vertically opposite angles)

And, (corresponding angles)

Therefore,

(Vertically opposite angles)

Again, (corresponding angles)

Hence, and .

(iii)

Statement: Two lines perpendicular to the same line are perpendicular to each other.

False

Reason:

The figure can be drawn as follows:

Here, and

It is given that , therefore,

(i)

Similarly, we have , therefore,

(ii)

From (i) and (ii), we get:

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

(iv)

Statement: Two lines parallel to the same line are parallel to each other.

True

Reason:

The figure is given as follows:

It is given that and

We need to show that

We have , thus, corresponding angles should be equal.

That is,

Similarly,

Therefore,

But these are the pair of corresponding angles.

Therefore, .

(v)

Statement: If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are equal.

False

Reason:

Theorem states: If a transversal intersects two parallel lines then the pair of alternate interior angles is equal.