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IF one Right Line DE cut or pass thro' another AB (Fig. 12) the opposite Angles at the top or intersection ACD and ECB are called Vertical; as also the other two ACE and DCB: Whence follow these
I. THat the Vertical Angles are always(a) 1.1 Equal; for both ACD and ECB with the third, ACE, which is common to both, fill or are equal to a Semicircle; as likewise both ACE and DCB with the third ECB, which is common.
II. Contrarywise, if at(a) 1.2 the Point C of the Right Line DE, the 2 opposite Lines AC and CB make the Vertical An∣gles x and z equal, then will AC and CB make one Right Line; for, since x and o make a Semicircle, and z and x are equal, by Hypoth. o and z will also make or fill a Semicircle, whose Diameter will be ACB.
III. By the same Argument it will appear, that of 4 Lines(b) 1.3 proceeding from the same Point so as to make the opposite Vertical Angles equal, the 2 oppo∣site ones AC and CB, as also DC and CE, will make each but one Right Line; for since all the 4 Angles together make a whole Circle, or 4 Right Angles, and the sum of x and o is equal (by Hypoth.) to the sum of o and z, it follows, that both the one and the other will make Se∣micircles, whose Diameter will be AB and DE, and so Right Lines.