Theorem 6.1: If two parallel lines are transected by a third, the alternate interior angles are the same size.
Proof: Let A and B denote the points where the transverse line meets the parallel lines. Let C be the midpoint of the line segment between A and B. Let DE be the line through C which is perpendicular to one, and hence, by Theorem 1.14, both of the parallel lines.
Let D be the point where the perpendicular line meets the line which contains the point A and let E be the point where the perpendicular line meets the line that contains the point B. If we rotate the figure 180o about point C , then point D is moved to point E and vice versa. The size of the angles between DE and the parallel lines are preserved by Theorem 5.6, so DE remains perpendicular to both of the lines, so, by Theorem 1.15, they remain parallel. Moreover, A and B are switched, so angle CAD is moved to angle CBE. Since a rotation is an invertible isometry by Theorem 5.13, it preserves the size of the angles by Theorem 5.6. We can conclude that the alternate interior angles are the same size.