Euclid's Elements Book I Proposition 28
If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.
In other words, when given two straight lines and a line that intersects those two lines:
1) If that line creates two corresponding angles that are equal to each other
OR
2) If that line creates two same-side interior angles that add up to 180 degrees
then the two straight lines will be parallel to one another.
PRE-REQUISITE KNOWLEDGE NEEDED:
Proposition 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.
Proposition 27: If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.
1) We create two straight lines AB and CD and create line EF that falls on those two straight lines.
2) First, we operate under the assumption that two corresponding angles, say angle EGB (labeled as alpha) and GHD (labeled as beta), are equal to each other.
3) We then can find that angle AGH (labeled as delta) is equal to angle EGB through Proposition 15.
4) Through this, we can see that angle AGH is also equal to angle GHD, as angle AGH is equal to angle EGB is equal to angle GHD.
5) Since angles AGH and GHD are alternate and equal to each other, we can find through Proposition 27 that lines AB and CD are parallel
1) Once again, we create two straight lines AB and CD and create line EF that falls on those two lines.
2) This time, we operate under the assumption that angles BGH (labeled as alpha) and GHD (labeled as beta) are equal to two right angles.
3) Using Proposition 13, we also have that angles AGH (labeled as delta) and BGH are equal to two right angles.
4) If we remove the common angle BGH, we then have that angles AGH and GHD are equal to each other.
5) Thus, according to Proposition 27, we have that lines AB and CD are parallel as angles AGH and GHD are equal to each other.
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