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Euclid: Elementa

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Proposition 13.
THEOR. 6. PROPOS. 13.
If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.
CVM recta linea super rectam consistens lineam angulos facit, aut duos rectos, aut duobus rectis æquales efficiet.

For let any straight line AB set up on the straight line CD make the angles CBA, ABD;  I say that the angles CBA, ABD are either two right angles or equal to two right angles.
RECTA linea AB, consistens super rectam CD, faciat duos angulos A B C, A B D.
Now, if the angle CBA is equal to the angle ABD, they are two right angles. [Def. 10]  But, if not, let BE be drawn from the point B at right angles to CD; [I. 11]  therefore the angles CBE, EBD are two right angles.  Then, since the angle CBE is equal to the two angles CBA, ABE, let the angle EBD be added to each;  therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. [C. N. 2]  Again, since the angle DBA is equal to the two angles DBE, EBA, let the angle ABC be added to each;  therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC. [C. N. 2]  But the angles CBE, EBD were also proved equal to the same three angles;  and things which are equal to the same thing are also equal to one another; [C. N. 1]  therefore the angles CBE, EBD are also equal to the angles DBA, ABC.  But the angles CBE, EBD are two right angles;  therefore the angles DBA, ABC are also equal to two right angles.
Si igitur A B, fuerit perpendicularis ad C D, erunt dicti 10. def anguli duo recti.   Si vero A B, non fuerit perpendicularis, faciet unum quidem angulum obtusum, alterum vero acutum. Dico igitur ipsos duobus esse rectis æquales. Educatur enim B E, ex B, perpendicularis ad C D,   ut sint duo anguli E B C, E B D, recti.  Quoniam vero angulus rectus E B D, æqualis est duobus angulis D B A, A B E; erunt, apposito communi angulo recto E B C,  duo recti E B D, E B C, tribus angulis D B A, A B E, E B C, æquales.   Rursus quia angulus A B C, duobus angulis A B E, E B C, æqualis est; apposito communi angulo A B D,  erunt duo anguli A B C, A B D, tribus angulis D B A, A B E, E B C, æquales.   Sed illis tribus ostensum fuit esse etiam æquales duos rectos E B D, E B C;  quæ autem eidem æqualia, inter se sunt æqualia.  Duo igitur anguli A B C, A B D, æquales sunt duobus rectis E B D, E B C.
Therefore etc.  Q. E. D.
Cum ergo recta linea super rectam consistens lineam, &c.   Quod ostendere oportebat.

SCHOLION
VIDETVR hæc propositio pendere ex communi quadam animi notione. Quo enim angulus A B C, superat rectum angulum E B C, eo reliquus angulus A B D, superatur ab angulo recto E B D. Nam sicut ibi excessus est angulus A B E, ita hic defectus est idem angulus A B E. Quocirca anguli A B C, A B D, duobus rectis æquales esse convincuntur, siquidem tantum unus eorum supra rectum acquirit, quantum alter deperdit.

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