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. Dem.—If the three lines be coplanar, the Proposition is evidently the same as I. xxx. If they are not coplanar, from any point G in EF draw in the planes of EF, AB; EF, CD, respectively, the lines GH, GK each perpendicular to EF [I. xi.]. Then because EF is perpendicular to each of the lines GH, GK, it is normal to their plane [XI. iv.]. And because AB is parallel to EF (hyp.), and EF is normal to the plane GHK, AB is normal to the plane GHK [XI. viii.]. In like manner CD is normal to the plane HGK. Hence, since AB and CD are normals to the same plane, they are parallel to one another. PROP. X.—Theorem. If two intersecting right lines (AB, BC) be respectively parallel to two other intersecting right lines (DE, EF), the angle (ABC) between the former is equal to the angle (DEF) between the latter.