Why are lines and parallel to each other?

Question

Homi · Accepted Answer

Understand the Question

If two lines in a plane are both perpendicular to the same line (transversal), they are parallel to each other.
Given that line $l$ is perpendicular to line $t$ ( $t \perp l$ ) and line $m$ is also perpendicular to line $t$ ( $t \perp m$ ), we can conclude that line $l$ is parallel to line $m$ ( $l \parallel m$ ).

Step 1 · Relationship Between Line $l$ and Line $t$

From the figure, the vertical line $t$ intersects the horizontal line $l$ at a right angle ( $90^\circ$ ). Diagram 1

Therefore $t \perp l$

Step 2 · Relationship Between Line $m$ and Line $t$

From the figure, the vertical line $t$ also intersects the horizontal line $m$ at a right angle ( $90^\circ$ ). Diagram 2

Therefore $t \perp m$

Step 3 · Determine the Relationship Between Lines $l$ and $m$

Since both lines $l$ and $m$ are perpendicular to the same line $t$ : $t \perp l \quad \text{and} \quad t \perp m$

Two lines that are perpendicular to the same line are parallel to each other.

Therefore $l \parallel m$

Answer

Lines $l$ and $m$ are parallel to each other ( $l \parallel m$ ) because both lines are perpendicular to the same line $t$ .

Common Mistakes

Symbol Confusion: Confusing the perpendicular symbol ( $\perp$ ) with the parallel symbol ( $\parallel$ ).
Angle Verification: Overlooking that corresponding angles formed with transversal line $t$ are both $90^\circ$ , which confirms that the lines must be parallel.

Question 17