Q1.
How do we implement this idea using a ruler and a compass?
Answer
Draw a transversal, then copy the corresponding angle at the new point. Equal corresponding angles force the lines to be parallel.
- Let m be the given line. Draw a line l crossing it at A. Line l is the transversal.
- Choose the point B on l through which the parallel must pass.
- With one radius, draw an arc from A cutting m at C and cutting l at D. With the same radius draw an arc from B, cutting l at E.
- Open the compass to CD and, from E, cut the arc at F.
- Join B to F and extend — this line n satisfies m ∥ n.
Angle a = angle between m and l at A
The copy at B is the corresponding angle for the transversal l
Corresponding angles equal ⇒ m ∥ n
The copy at B is the corresponding angle for the transversal l
Corresponding angles equal ⇒ m ∥ n
Why it happens: two lines cut by a transversal are parallel exactly when a pair of corresponding angles is equal. We cannot check "never meeting" by drawing, since we can only draw a finite bit of a line. But we can guarantee the angle, and the angle guarantees the parallelism.
Tip: when you used a ruler and set square in Grade 6, you were doing the same thing — the set square slid along, carrying a fixed angle. Here the compass carries the angle instead.