State whether the following statement is true or false.

Two lines are perpendicular to one another if they are both parallel to the same line.

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True

False

answer is B.

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Detailed Solution

It is given that two lines are perpendicular to another line.
We know that a transversal line intersects two other lines at two different points within the same plane. The angles in corresponding corners on a transversal are known as corresponding angles.

∠ 1 & ∠ 5

and

∠ 2 & ∠ 6

in the following diagram are examples of corresponding angles.
Question Image

The two lines will be parallel if the corresponding angles are equal, and corresponding angles will also be equal if the lines are parallel. Co-interior angles are those that are located between two lines and on the same transverse sides of a line.
Co-interior angles in the above figure are

∠ 3 & ∠ 6

and

∠ 4 & ∠ 5.

The two lines will be parallel if the sum of the co-interior angles is

180 °,

and vice versa if the lines are parallel.
Question Image

Let’s assume that the lines l || m and l || n and t is a transversal.
Using the corresponding angles property, we have:

∠ 1 = ∠ 2

and ∠ 2 = ∠ 3 ⇒ ∠ 1 = ∠ 3

Thus, if the transversal line t intersects three lines ‘l’ ‘m’, and ‘n’ such that l || m and l || n, then m || n.
Therefore, two lines are parallel to one another if they are both parallel to the same line.
Thus, the given statement is not true.
Hence, option (2) is correct.

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