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THe Lines which are perpendicular to the same Plane, are parallel.
If the Lines AB, CD, be perpendi∣cular to the same plane EF; they shall be parallel. It is evident that the inter∣nal Angles ABD, BDC, are right; but that is not sufficient, for we must also prove that the Lines AB, CD, are in the same Plane. Draw DG, perpendi∣cular to BD, and equal to AB: Draw also the Lines BG, AG, AD.
Demonstration. The Triangles ABD, BDG, have the sides AB, DG, equal: BD is common; the Angle•• ABD, BDG, are right. Thence the Bases AD, BG, are equal (by the 4th. of the 1st.) Moreover the Triangles ABG, ADG, have all their sides equal; thence the Angles ABG, ADG, are equal; and ABG being right, seeing AB is perpendicular to the Plane, the Angle ADG is right. Therefore the Line DG is perpendicular to the three Lines CD, DA, DB, which by consequence are in the same Plane (by the 5th.) Now the Line AB is also in the plane of the Lines AD, DC, (by the second;) thence AB, CD, are in the same plane.
Coroll. Two Lines which are parallel are in the same Plane.
WE demonstrate by this Proposition, that the hour Lines are parallels amongst themselves, in all Planes which are parallel to the Axis of the World; as in the Polar and Meridian Dials, and others.