Parallel through a point, step by step

Construct a line through a point parallel to a given one.

The question

Construct the parallel through P.

compass+straightedge

Step 1→Draw any line through P that crosses the given line. Call the crossing X

Step 2→Copy the angle at X onto the new line, with its vertex at P

Step 3→The arm you draw is parallel to the given line

Step 4→The two angles are corresponding angles, which is why the lines cannot meet

arcs=the evidence

Construct the parallel through P. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
1. Draw any line through P that crosses the given line. Call the crossing X. Any line through P that crosses the given one will do. It is there to create an angle you can copy, not because its direction matters.
2. Copy the angle at X onto the new line, with its vertex at P. Copying the angle at P puts the same corresponding angle in the same position on the transversal.
3. The arm you draw is parallel to the given line. Equal corresponding angles force the two lines apart forever — that is the converse of the corresponding-angles theorem, and it is why this construction is a proof rather than a good guess.
4. The two angles are corresponding angles, which is why the lines cannot meet. The arm you drew is therefore parallel to the given line, exactly, without ever measuring an angle.
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.

How it works.

01
Construct the parallel through P. Two tools only — a compass that holds a distance and a straightedge that draws a line. Nothing here is measured.
02
1. Draw any line through P that crosses the given line. Call the crossing X. Any line through P that crosses the given one will do. It is there to create an angle you can copy, not because its direction matters.
03
2. Copy the angle at X onto the new line, with its vertex at P. Copying the angle at P puts the same corresponding angle in the same position on the transversal.
04
3. The arm you draw is parallel to the given line. Equal corresponding angles force the two lines apart forever — that is the converse of the corresponding-angles theorem, and it is why this construction is a proof rather than a good guess.
05
4. The two angles are corresponding angles, which is why the lines cannot meet. The arm you drew is therefore parallel to the given line, exactly, without ever measuring an angle.
06
Leave every arc showing. The arcs are the evidence the figure was constructed rather than eyeballed, and a marker reads them to check the method as well as the result.