First Method

Draw the straight line A C about 3 1/2 inches long and assume the point P about 1 1/2 inches above A C. Through the point P draw an oblique line F E forming any convenient angle - about 60 degrees - with A C. Now construct an angle equal to P F C having its vertex at P and the line E P as one side (See Problem 2.) The straight line P 0 forming the other side of the angle EPO will be parallel to A C.

Proof

If two straight lines are cut by a third making the corresponding angles equal, the lines are parallel.

Second Method

Draw the straight line A C about 3 1/4 inches long and assume the point P about 1 1/2 inches above A C. With P as a center and any convenient radius - about 2 1/2 inches - draw the indefinite arc E D cutting the line A C. Now with the same radius and with D as a center, draw an arc P Q. Set the compass so that the distance between the needle point and the pencil is equal to the chord P Q. With D as a center and a radius equal to P Q, describe an arc cutting the arc E D at H. A line drawn through P and H will be parallel to A C.

Proof

Draw the line Q H. Since the arcs P Q and H D are equal and have the same radii, the angles P H Q and H Q D arc equal. Two lines are parallel if the alternate interior angles arc equal.