Why are lines and parallel to each other?

Question

Homi · Accepted Answer

We need to understand the relationship between lines $l$ and $m$ .

Step 1 — Observe the relationship between line $l$ and line $t$

Look at the third diagram in the figure. Line $l$ is a dashed horizontal line. Line $t$ is a dashed vertical line. They cross each other. There is a small square symbol where they cross. This symbol means they meet at a right angle. So, line $t$ is perpendicular to line $l$ .

$t \perp l$

Diagram 1

Step 2 — Observe the relationship between line $m$ and line $t$

Now look at the fourth diagram in the figure. Line $m$ is a dashed horizontal line. Line $t$ is the same dashed vertical line. They also cross each other. There is a small square symbol where they cross. This symbol means they meet at a right angle. So, line $t$ is perpendicular to line $m$ .

$t \perp m$

Diagram 2

Step 3 — Conclude about lines $l$ and $m$

From Step 1, we know that line $t$ is perpendicular to line $l$ . From Step 2, we know that line $t$ is perpendicular to line $m$ . Both lines $l$ and $m$ are perpendicular to the same line $t$ . There is a rule in geometry for this situation. If two lines are perpendicular to the same line, then they must be parallel to each other. Therefore, line $l$ is parallel to line $m$ .

$\boxed{l \parallel m}$

Answer

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

Question 17

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