Sports day helpers are marking a second running lane line through point \(Q\) that must stay exactly parallel to the existing lane line \(NP\), so every run...

Sports day helpers are marking a second running lane line through point \(Q\) that must stay exactly parallel to the existing lane line \(NP\), so every runner has the same width of lane. A line through Q is parallel to \(NP\) only if it never changes distance from \(NP\), which is guaranteed by making the angle it forms with a transversal equal to the angle that \(NP\) forms with the same transversal (equal corresponding angles).

1. Draw a straight line from Q down to a point on \(NP\); this transversal crosses \(NP\) at some angle. With the compass point on that crossing point, draw an arc of radius \(r\) that cuts both the transversal and \(NP\), then measure the distance between where it cuts them. [1]
2. With the same radius \(r\), draw an arc centred on Q cutting the transversal (extended beyond Q). Open the compasses to the distance measured in the first step and, from where that arc cuts the transversal, draw a further arc to fix a second point on the first arc. The line through Q and this new point reproduces the same angle with the transversal, so it never meets \(NP\) and is the required parallel line. [1]

The two radii used to copy the angle must match exactly between the two vertices; changing the compass setting partway through the construction breaks the equal-angle property that makes the line parallel.