The Elements, Book I –
Propositions 22 – 28
MONT 104Q – Mathematical
Journeys: Known to Unknown
October 2, 2015
Constructing triangles and angles
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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
a) One side and the two adjacent angles of one triangle are equal
to one side and the two adjacent angles of the other triangle
b) 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.
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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
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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
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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
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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
a)
Two corresponding angles are congruent, or
b)
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