Parallelism passes along
Line l is parallel to m. Line n is parallel to m. Is l parallel to n?
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Step 1 of 4
- Lines l, m and n, cut by one transversal.
- l and m: the corresponding angle is 60°.
- m and n: the corresponding angle is the same 60°.
- So l and n also have equal corresponding angles.
The rule
Yes, when all three lines are on one flat page.
Theorem. Three lines lie on one flat page. Two of them are parallel to the third. Then they are parallel to each other.
Proof. Let l and n both be parallel to m. Draw one transversal across all three lines. Mark the corresponding angles 1, 2 and 3. Angle 1 is on l. Angle 2 is on m. Angle 3 is on n.
- l is parallel to m. So angle 1 = angle 2.
- n is parallel to m. So angle 3 = angle 2.
- Both equal angle 2. So angle 1 = angle 3.
- Angles 1 and 3 are corresponding angles for l and n. Equal corresponding angles make l parallel to n.
The proof goes through m. Neither given fact compares l with n directly.