## Euclid, Elements of Geometry, Book I, Proposition 13 (Edited by Dionysius Lardner, 1855)

Proposition XIII. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]

 (82) When a right line (A B) standing upon another (D C) makes angles with it, they are either two right angles, or together equal to two right angles.

If the right line A B is perpendicular to D C, the angles A B C and A B D are right (11). If not, draw B E perpendicular to D C (XI), and it is evident that the angles C B A and A B D together are equal to the angles C B E and E B D, and therefore to two right angles.

The words ‘makes angles with it,’ are introduced to exclude the case in which the line A B is at the extremity of B C.

(83)   From this proposition it appears, that if several right lines stand on the same right line at the same point, and make angles with it, all the angles taken together are equal to two right angles.

Also if two right lines intersecting one another make angles, these angles taken together are equal to four right angles.

The lines which bisect the adjacent angles A B C and A B D are at right angles; for the angle under these lines is evidently half the sum of the angles A B C and A B D.

If several right lines diverge from the same point, the angles into which they divide the surrounding space are together equal to four right angles.

(84)   When two angles as A B C and A B D are togther equal to two right angles, they are said to be supplemental, and one is called the supplement of the other.

(85)   If two angles as C B A and E B A are together equal to a right angle, they are said to be complemental, one one is said to be the complement of the other.

Next: Proposition 14

Previous: Proposition 12

This proposition in other editions: