Parallel through a point

The question

Construct the parallel through P.

How it works.

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

01
Construct the parallel through P. Use only a compass and a straightedge — no measuring.
02
1. Draw any line through P that crosses the given line. Call the crossing X. Its direction does not matter; it only gives you an angle to copy.
03
2. Copy the angle at X onto the new line, with its vertex at P. The copied angle sits in the same position on this line as the original, so the two are corresponding angles.
04
3. The arm you draw is parallel to the given line. Equal corresponding angles mean the lines are parallel.
05
4. The two angles are corresponding angles, which is why the lines cannot meet. No angle was measured, so the lines are parallel exactly, not nearly.
06
Leave every arc showing — they show how the figure was made.