Bitcoin is Converses of Props 27 and 28 | Euclid's Elements Book 1 Prop 29

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"alternate angles equal to one another" -> Green = Pink
"exterior angle" -> Red
"exterior angle equal to the interior and opposite angle" -> Red = Pink
"sum of the interior angles on the same side" -> Pink + Blue = 180°

Arg 1:

- Suppose Green > Pink
- Then Green + Blue > Pink + Blue
- Green + Blue = 2 right angles because line
- Therefore Pink + Blue < 2 right angles
- By the parallel postulate, the two lines AB and CD meet in the pink/blue direction
- We assumed the lines were parallel
- Contradiction
- Therefore Green = Pink

Arg 2:

- Green = Red because vertical angles
- Therefore Red = Pink
- Red + Blue = Pink + Blue
- Red + Blue = 2 right angles because straight line
- Therefore Pink + Blue = 2 right angles

Parallel postulate: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/post5.html
Element page: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI29.html
GeoGebra: https://www.geogebra.org/m/squyduwp
Universal Hyperbolic Geometry: https://youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI

Euclid Elements playlist: https://odysee.com/@DoctorAjayKumar/ee
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Euclid Elements website: https://mathcs.clarku.edu/~djoyce/java/elements/

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