1. Book 1, Proposition 29 By: Isabel Block and Alexander Clark
Euclid’s Elements:
Book 1, Proposition 29
By: Isabel Block and Alexander Clark

2. What are we trying to prove?
A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.

3. Construct Create parallel lines AB and CD
Construct line EF so that it intersects AB and CD at points G and H
This should create alternate interior angles AGH and GHD
This should create alternate exterior angles EGB and FHC A B C D E F G H

4. Step 1 Say <AGH ≠ <GHD (Common Notion 6)
That means one is greater
Say <AGH > <GHD
<AGH + <BGH > <GHD + <BGH (Common Notion 6)
<AGH + <BGH = 2 right angles (Proposition 13)
<BGH + < GHD < 2 right angles (Proposition 13)
Contradiction: (Postulate 5)
AB would have to meet with CD at some point, making the lines not parallel
Hypothesis says they are parallel

5. Step 2 ∴ <AGH = <GHD <AGH = <EGB (Proposition 15)
<EGB = <GHD (Common Notion 1)
<EGB + <BGH = <GHD + <BGH (Common Notion 2)
<EGB + <BGH = 2 right angles (Proposition 13)
∴ <BGH + <GHD = 2 right angles (Common Notion 1)

6. Relationship to Proposition 27
Proposition 29 is the converse to Proposition 27
27 says that if the alternate interior angles are equal (when line EF intersects AB and BC), then the lines AB and BC are parallel
However, the converse cannot be assumed to be true
29 assumes that the alternate interior angles are not equal, proving that the lines would not be parallel
By hypothesis, the lines are parallel in 29, so this is a contradiction

7. Importance This is the first time Euclid uses Postulate 5
#1-4 are the “absolute geometry” postulates which he uses for propositions 1-28
Postulate 5 cannot be proven as a theorem
Used to prove later propositions (ex. Proposition 30 and 32) in Book I
Also used frequently in Books II, VI, and XI, and once in Book XII
Euclidean geometry vs. elliptic geometry vs. hyperbolic geometry
Distinguished by the number of lines parallel to a given line passing through a given point
Euclidean: 1
Elliptic: 0
Hyperbolic: ∞
This is the first proposition that does not hold in hyperbolic geometry