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.
Under Development
Last updated on: 2022-04-26.
Commentary
1. Let AB and PB be two lines standing on a given straight line CD , with B being the foot.
2. LetAB 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.
2. Let
3.
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.
