Question

Gerald is constructing a line parallel to line l through point P. He begins by drawing line m through points P and Q. He then draws a circle centered at Q, which intersects line l at point N and line m at point S. Keeping the compass measure, he draws a congruent circle centered at point P, which intersects line m at point T.

Which next step will create point R, such that when a line is drawn through points P and R, the line will be parallel to line l?

Lines m and n intersect at point Q. A circle is drawn around point Q and forms point S on line m and forms point N on line l. Point P is also on line m. A circle is drawn around point P and forms point T on line m.

Use the compass to construct a circle centered at Q through point P.
Using the compass measure between points S and N, draw an arc to the right of line m, centered at T, intersecting the edge of circle P.
Using the compass measure between points S and N, draw an arc above line l, centered at N, intersecting the edge of circle Q.
Use the compass to construct a circle centered at P through point Q.

Answer 1

"Using the compass measure between points S and N, draw an arc to the right of line m, centered at T, intersecting the edge of circle P."

Explanation (brief): Circles centered at Q and P are congruent, so S corresponds to T. Copying the chord SN as an arc centered at T produces R on the circle around P so that the arc TR subtends the same central angle at P that SN subtends at Q. That makes the angle between m and PR equal to the angle between m and l, so PR is parallel to l.