The Elements, Book I – Propositions 22 – 28 MONT 104Q – Mathematical Journeys: Known to Unknown October 2, 2015
Constructing triangles and angles Proposition 22. To construct a triangle if the three sides are given. The idea should be clear – given one side, find the third corner by intersecting two circles (Postulate 3). This only works if the statement of Proposition 20 (the ``triangle ineqality'') holds. Proposition 23. To construct with a given ray as a side an angle that is congruent to a given angle This is based on finding a triangle with the given angle (connecting suitable points using Postulate 2), then applying Proposition 22.
Propositions 24 and 25 Proposition 24. If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base. Proposition 25. If two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other. These statements seem most closely related to the SAS congruence criterion from Proposition 4. But they are not used in the rest of Book I, so we'll skip over them.
Additional triangle congruences Proposition 26. Two triangles are congruent if One side and the two adjacent angles of one triangle are equal to one side and the two adjacent angles of the other triangle One side, one adjacent angle, and the opposite angle of one triangle are equal to one side, one adjacent angle, and the opposite angle of the other triangle. Both statements here are cases of the “AAS” congruence criterion as usually taught today in high school geometry. The proof actually relies on the “SAS” statement from Proposition 4.
Theory of parallels Proposition 27. If two lines are intersected by a third line so that the alternate interior angles are congruent, then the two lines are parallel. As for us, parallel lines for Euclid are lines that, even if produced indefinitely, never intersect Say the two lines are AB and CD and the third line is EF as in the following diagram
Proposition 27, continued The claim is that if <AEF = <DFE, then the lines AB and CD, even if extended indefinitely, never intersect. Proof: Suppose they did intersect at some point G
Proposition 27, concluded Then the exterior angle <AEF is equal to the opposite interior angle <EFG in the triangle ᐃEFG. But that contradicts Proposition 16. Therefore there can be no such point G. QED
Parallel criteria Proposition 28. If two lines AB and CD are cut by third line EF, then AB and CD are parallel if either Two corresponding angles are congruent, or Two of the interior angles on the same side of the transversal sum to two right angles.
Parallel criteria Proof: (a) Suppose for instance that <GEB = <GFD. By Proposition 15, <GEB = <AEH. So <GFD = <AEH (Common Notion 1). Hence AB and CD are parallel by Proposition 27.
Parallel criteria Proof: (b) Now suppose for instance that <HEB + <GFD = 2 right angles. We also have <HEB + < HEA = 2 right angles by Proposition 13. Hence <GFD = <HEA (Common Notion 3). Therefore AB and CD are parallel by Proposition 27. QED