5. Two right lines passing through a point equidistant from two parallels intercept equal portions on the parallels.
6. The perimeter of the parallelogram, formed by drawing parallels to two sides of an equilateral triangle from any point in the third side, is equal to twice the side.
7. If the opposite sides of a hexagon be equal and parallel, its diagonals are concurrent.
8. If two intersecting right lines be respectively parallel to two others, the angle between the former is equal to the angle between the latter. For if AB, AC be respectively parallel to DE, DF, and if AC, DE meet in G, the angles A, D are each equal to G [xxix.].
PROP. XXX.—Theorem.
If two right lines (AB, CD) be parallel to the same right line (EF), they are parallel to one another.
Dem.—Draw any secant GHK. Then since AB and EF are parallel, the angle AGH is equal to GHF [xxix.]. In like manner the angle GHF is equal to HKD [xxix.]. Therefore the angle AGK is equal to the angle GKD (Axiom i.). Hence [xxvii.] AB is parallel to CD.
PROP. XXXI.—Problem.
Through a given point (C) to draw a right line parallel to a given right line.
Sol.—Take any point D in AB. Join CD (Post. i.), and make the angle DCF equal to the angle ADC [xxiii.]. The line CE is parallel to AB [xxvii.].