Final answer:
To construct parallels to a given line through specified points, one uses a compass and straightedge to create equidistant arcs that define points through which a straight line is drawn. The resulting lines are parallel to the original line and to each other.
Step-by-step explanation:
To construct a parallel to line l through a point P, you would perform the following steps:
- Place the compass at point P and draw an arc that intersects line l at two points. Label these intersection points A and B.
- Without changing the compass width, place the compass at point A and draw an arc above the line. Repeat this step with the compass at point B.
- Label the intersection of the two arcs as point C.
- Using a straightedge, draw a line through point P and point C. This is the parallel line through P.
To construct parallels through A and B you would follow similar steps for each point, ensuring the spacing of the compass is identical to the spacing used when drawing the first parallel line. This will ensure that the constructed lines are equidistant and parallel to line l.
The relationship between the two lines constructed through A and B is that they are parallel to each other as they are both equidistant from and parallel to the original line l.