Question

Consider the following statements relating to 3 lines $L_1$, $L_2$ and $L_3$ in the same plane
(1). If $L_2$ and $L_3$ are both parallel to $L_1$, then they are parallel to each other.
(2). If $L_2$ and $L_3$ are both perpendicular to $L_1$, then they are parallel to each other.
(3). If the acute angle between $L_1$ and $L_2$ is equal to to acute angle between $L_1$ and $L_3$, then $L_2$ is parallel to $L_3$.

Of these statements:

• A. (1) and (2) are correct
• B. (1) and (3) are correct
• C. (2) and (3) are correct
• D. (1), (2) and (3) are correct

Answer 1

The correct option is A (1) and (2) are correct

(1) If L₂ and L₃ are both parallel to L₁, then they are parallel to each other.
This is true. We can prove this using contradiction, i.e., let L₂ and L₃ are not parallel.

Now, we know that if two lines in a plane are not parallel, then they intersect each other at some point.
Therefore, L₂ and L₃ intesect each other at some point.
Since, L₁ is parallel to L₂, then L₁ must also intesect L₃. But, we have already assumed L₁ and L₃ are parallel.

This implies that, our assumption that L₂ and L₃ are not parallel is incorrect.
Hence, L₂ and L₃ are parallel to each other.

(2) If L₂ and L₃ are both parallel to L₁, then they are parallel to each other.
This is true. We can prove this using contradiction, i.e., let L₂ and L₃ are not parallel.

If L₂ and L₃ are not parallel, implies that they intersect at some point. Let this point be A.

Also, L₂ and L₃ are both perpendicular to L₁, implies that L₂ and L₃ intersect L₁ at some point. Let these points be B and C respectively.

Now, we know that sum of angles of a triangle is 180°. We can use this fact to calculate the angle ∠BAC.
∠BAC + ∠ABC + ∠ACB = 180°
Since, L₂ and L₃ are both perpendicular to L₁,
∠ABC = 90° and ∠ACB = 90°.
Using the above fact, we can determine the angle between L₂ and L₃, i.e,
∠BAC = 180° − 90° − 90°
∠BAC = 0°

This contradicts our assumption that L₂ and L₃ are not parallel.
Hence, L₂ and L₃ are parallel to each other.

(3) If the acute angle between L₁ and L₂ is equal to to acute angle between L₁ and L₃, then L₂ is parallel to L₃.
This is not true. We can prove this by giving a counter example.
Let the angle between L₁ and L₂ be 30°. This implies that, angle between L₁ and L₃ will also be 30°.
We can draw such lines in such a way that L₁ is a angle bisector of the angle between L₂ and L₃. This gives us the angle between L₂ and L₃ as 60°.
This shows that the given statement is not true always.