Final answer:
To construct a parallel line using a compass and straightedge, first draw a line with a point not on it. Copy an angle from the initial line and translate it to the point not on the line. Then draw a line through the translated angle's intersection point to create a parallel line.
Step-by-step explanation:
To construct a second line through a point that is parallel to a given line using a compass and straightedge, follow these steps:
Draw a line and mark a point not on the line.
Choose a point on the existing line and use the compass to copy the angle formed by the line and a horizontal line passing through the chosen point.
Place the compass' pointed end on the marked point not on the line, and draw an arc.
Without changing the compass' width, replicate this arc intersection with your previously drawn arc. This will ensure that the angles are congruent.
Use a straightedge to draw a line through the point and the intersection of the arcs, which creates a line that is parallel to the original line.
This method is known as duplicating corresponding angles, and the key concept here is that parallel lines have corresponding angles that are equal, which is a postulate in Euclidean geometry. It is through the process of duplicating the angle that we assure parallelism.