Euclid, Elements I 13©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 13: If a straight-line stood against a straight-line makes angles, it will make either two right-angles or equals to two right angles.

(general diagram)

Diagrams for book 1 proposition 13, following the display, construction, and demonstration(diagram 1) For let a certain straight-line, AB, stood on a straight-line, GD, make angles, those by GBA, ABD. I say that the angles by GBA, ABD either are two right-angles or are equal to two right-angles.
(diagram 2) And so, if the angle by GBA is equal to that by ABD, they are two right angles. (diagram 3=1) But if not, let BE be drawn from point B to straight-line GD at right angles. (diagram 4) Therefore, the angles by GBE, EBD are two right angles. (diagram 5) And since the angle by GBE is equal to two, those by GBA, ABE, (diagram 6) let a common be added, that by EBD. Therefore, the angles by GBE, EBD are equal to three, those by GBA, ABE, EBD. (diagram 7) Again, since the angle by DBA is equal to two, those by DBE, EBA, (diagram 8) let a common be added, that by ABG. Therefore, the angles by DBA, ABG are equal to three, those by DBE, EBA, ABG. (diagram 9) But the angles by GBE, EBD were also shown equal to three, the same ones. But those equal to the same are also equal to one another. Therefore, the angles by GBE, EBD are also equal to those by DBA, ABG. But the angles by GBE, EBD are two right angles. Therefore, the angles by DBA, ABG are equal to two right angles. Therefore, if a straight-line stood against a straight-line makes angles, it will make either two right-angles or equals to two right angles, just what it was required to show.

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