Lines Parallel to the Same Line

A small theorem with big consequences: if two lines are both parallel to a third line, then they are parallel to each other. Sometimes called the transitivity of parallelism, this fact lets us chain parallelism arguments and is invoked constantly in proofs about quadrilaterals, parallelograms, and beyond.

Definition

Transitivity of parallelism. If $\ell \parallel n$ and $m \parallel n$, then $\ell \parallel m$.

This is the parallelism analogue of "if $a = b$ and $b = c$ then $a = c$" , a transitive relation.

Proof

Proof (by contradiction). Suppose $\ell$ and $m$ are not parallel. Then they meet at some point $P$. Now from $P$, we have two lines ($\ell$ and $m$) each parallel to the third line $n$. But by Playfair's axiom (a form of the parallel postulate), through a point not on $n$ there is exactly one line parallel to $n$. So $\ell$ and $m$ must be the same line , contradicting our assumption that they meet at one point as two distinct lines.

Hence $\ell$ and $m$ do not meet , they are parallel. Q.E.D.

The proof relies on Playfair's axiom (Euclid's fifth postulate in modern dress). Without the parallel postulate, the theorem would fail.

A natural extension

The same idea extends: if three or more lines are all parallel to a common line, they are all parallel to each other. In a figure with many parallel lines (like the rules of a notebook page), every pair of them is parallel.

Two practical uses

Use 1: proving parallelism by an intermediate line. Given $\ell$ and $m$ that are hard to compare directly, find a line $n$ such that both are parallel to $n$. Then $\ell \parallel m$.

Use 2: combining with the converse of the alternate-interior theorem. If you can show that a transversal makes equal alternate-interior angles with $\ell$ and $n$, and also with $m$ and $n$, then both $\ell \parallel n$ and $m \parallel n$, hence $\ell \parallel m$.

Worked examples

Example 1. Lines $p, q, r, s$ are all parallel to a single line $t$. State the parallelism between $p$ and $r$.

By the transitivity of parallelism, $p \parallel r$.

Example 2. Three lines $a, b, c$ are such that $a \parallel b$ and $b \parallel c$. Are $a$ and $c$ parallel?

Yes. By the transitivity theorem, $a \parallel c$.

Example 3. In a figure, three lines are drawn on a page. A transversal makes corresponding angles of $60^\circ$ with the first two lines, and $60^\circ$ with the second and third lines. Are the first and third lines parallel?

Yes. By the converse of the corresponding-angles theorem, line 1 is parallel to line 2, and line 2 is parallel to line 3. By transitivity, line 1 is parallel to line 3.

Example 4. Lines $\ell$, $m$, $n$ are such that $\ell \parallel m$ and $\ell \parallel n$, but $m$ and $n$ are not the same line. Are $m$ and $n$ parallel?

Yes, by transitivity: $m \parallel \ell$ and $\ell \parallel n$ implies $m \parallel n$.

Example 5. Prove that two distinct lines parallel to the same line cannot intersect.

This is exactly the transitivity theorem. If they intersected, we would have two distinct lines through the intersection point, each parallel to the same line , contradicting Playfair.

Try it yourself

1. State the transitivity of parallelism.
2. Prove it using Playfair's axiom (in your own words).
3. Lines $\ell, m, n$ are such that $\ell \parallel m$ and $n \parallel m$. Are $\ell$ and $n$ parallel?
4. Three lines are drawn so that the corresponding angles between successive pairs are all $50^\circ$. Are the first and third parallel?
5. Prove that lines $AB, CD, EF$ are parallel if $AB \parallel CD$ and $CD \parallel EF$.
6. If $\ell$ is parallel to $m$ and $m$ is parallel to $\ell$, can we say $\ell = m$?
7. Show that four parallel rules on a notebook are all parallel to each other.
8. Lines $a$ and $b$ are not parallel; line $c$ is parallel to $a$. Can $c$ also be parallel to $b$? Justify.
9. State whether "transitive" applies to parallelism in your own words.
10. Why is Playfair's axiom essential for this theorem?

Pitfalls / Insight

• Distinct lines. The theorem assumes the lines are distinct. If two of them happen to be the same line, "transitivity" is vacuously true.
• Playfair's axiom is the engine. Without the parallel postulate (or an equivalent), the theorem fails.
• Equally true for many lines. A whole family of lines parallel to one shared line are all parallel to each other.

Insight. Transitivity is a small theorem with broad reach. Whenever a problem involves multiple parallel lines, the theorem lets you "chain" them , proving parallelism via an intermediate parallel.