PROP. VII.—Theorem.
Two parallel lines (AB, CD) and any line (EF) intersecting them are coplanar.
Dem.—If possible let the intersecting line be out of the plane, as EGF. And in the plane, of the parallels draw [I. Post. ii.] the right line EHF. Then we have two right lines EGF, EHF, enclosing a space, which [I. Axiom x.] is impossible. Hence the two parallel right lines and the transversal are coplanar.
Or thus: Since the points E, F are in the plane of the parallels, the line joining these points is in that plane [I. Def. vi].
PROP. VIII.—Theorem.
If one (AB) of two parallel right lines (AB, CD), be normal to a plane (X), the other line (CD) shall be normal to the same plane.
Dem.—Let AB, CD meet the plane X in the points B, D. Join BD. Then the lines AB, BD, CD are coplanar. Now in the plane X, to which AB is normal, draw DE at right angles to BD. Take any point E in DE, and join BE, AE, AD.
Then because AB is normal to the plane X, it is perpendicular to the line BE in that plane [XI. Def. vi.]. Hence the angle ABE is right; therefore AE2 = AB2 + BE2 = AB2 + BD2 + DE2 (because BDE is right (const.)) = AD2 + DE2 (because ABD is right (hyp.)). Therefore the angle ADE is right. Hence DE is at right angles both to AD and BD. Therefore [XI. iv.] DE is perpendicular to CD, which is coplanar and concurrent with AD and BD. Again, since AB and CD are parallel, the sum of the angles ABD, BDC is two right angles [I. xxix.]; but ABD is right (hyp.); therefore BDC is right. Hence CD is perpendicular to the two lines DB, DE, and therefore [XI. iv.] it is normal to their plane, that is, it is normal to X.
PROP. IX—Theorem.
Two right lines (AB, CD) which are each parallel to a third line (EF) are parallel to one another.
