Constructions
140. The Parallel Path · equal angles, forever parallel
The corresponding angles ∠PQB and ∠QPR remain equal as P and Q move, keeping ST parallel to AB.
To construct a line parallel to AB through external point P: copy an angle ∠B from line AB at point P, on the same side. The new line creates equal corresponding angles with the transversal PQ, so it is parallel to AB.
What this lesson covers
Try to break it
Drag Q along line AB and P off the line. The transversal PQ shifts, and the corresponding angles change size together — but they stay perfectly matched. Because of that match, the line through P stays parallel to AB and the two lines refuse to cross.
How you build it
Draw line AB and mark external point P.
- Draw line AB across the canvas. This is the given line you will draw a parallel to.
- Mark point P above line AB. The parallel line will pass through P.
- Draw a line from P down through AB. This transversal meets AB at Q — the angle at Q is what you will copy at P.
- Mark point Q where the transversal crosses AB.
- Place the compass on Q and swing an arc that crosses AB to the right at C, and the transversal above Q at D.
- Mark point C where the arc crosses AB to the right of Q.
- Mark point D where the arc crosses the transversal above Q, between Q and P.
- Keep the same radius. Place the compass on P and swing an arc that crosses the transversal above P (the extension beyond P, away from Q). That crossing point is E.
- Mark point E where the arc from P crosses the extension of the transversal above P.
- Click C as the centre, then click D. This draws an arc at C passing through D and locks the compass width to the chord CD.
- Keep the same compass width (= chord CD). Place the compass on E and swing an arc. Where it crosses the arc from step 8 is point F.
- Mark point F where the two arcs intersect. This fixes the direction of the parallel line.
- Draw a line through P and F. This is the parallel line — ∠QPF equals ∠PQC, so PF is parallel to AB by corresponding angles.
The proof, step by step
Prove that the constructed line ST is parallel to AB.
- Given: Line AB and point P. Q is on AB. PQ is the transversal.
- Construction: ∠QPR is drawn equal to ∠PQB at point P.
- ∠PQB and ∠QPR are corresponding angles.
- Since corresponding angles are equal, line ST is parallel to line AB.
Worked example
In the figure, line AB || line ST. Transversal PQ intersects them at Q and P respectively. If ∠PQB = 72°, find the measure of ∠QPR.
Since AB || ST and PQ is a transversal, the corresponding angles ∠PQB and ∠QPR are equal. Therefore, ∠QPR = ∠PQB = 72°.
- 72° — correct
- 108°
- 90°
- 18°