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Computable Euclid

Proposition 13

Theorem

One line (AB  or PB ) standing on another line (CD ) will make either two right angles (ABC, ABD) or two adjacent angles (PBC, PBD) whose sum is equal to two right angles.

Commentary

1. Let AB  and PB  be two lines standing on a given straight line CD , with B being the foot.
2. Let AB  be perpendicular to CD  at point B. Then by definition of perpendicular (def. 10), the two adjacent angles ABC and ABD are both right angles.
3. PB  and CD  form two adjacent angles, PBC and PBD, which add up to two right angles.
4. Note that the measurement of a planar angle in degrees did not exist for Euclid. He considered the addition in a geometric way by adding shapes together, and here "equal" means putting two unequal adjacent angles together having the same form as putting two adjacent right angles together.
5. The following proposition, Book 1 Proposition 14, is the converse of this proposition.

Original statement

ἐὰν ϵὐθϵῖα ἐπ᾽ ϵὐθϵῖαν σταθϵῖσα γωνίας ποιῇ, ἤτοι δύο ὀρθὰς ἢ δυσὶν ὀρθαῖς ἴσας ποιήσϵι.

English translation

If a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles.


Computable version


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