Table of Contents
Aye, they had it tough...
Constructions
Construct a true square
Construct a pentagon
Construct tangents to a circle
Congruence and Equality
Old Fashioned Congruence
Uses of Congruence
Parallel Lines
Geometry::Algebra
Playfair’s Axiom
1.2 notebooks problems and hints
Problem 6, p. 18
Problem 8
In class problems
Theorem 1.2.1
The words “equivalent” force us to write
two proofs.
If E, then P.
Well, let’s put in a new line VW that is on p
and then explore what kind of angle or slope VW has.
Angle RPQ is a right angle , by construction. VW is
a straight line different from RS so one of the two angles VPQ or WPQ is strictly bigger
than 90 degrees -- I will match the labels and declare VPQ to be the obtuse one and OPQ to
be the acute one.
Angle PQU is 90 degrees, by construction.
If P, then E.
Since Playfair’s Axiom holds,
It is essential that the second requirement from
Euclid’s fifth postulate also be found to be true ---
Let’s go back to the sketch to do some
arithmetic and some arithmetic reasoning.
So angles DCB and CBA are part of a triangle and
the “LHS” pair is not.
About Theorems
Failure, if it isn’t lethal, brings growth.
Forging the tools to solve these problems was a
famous and exciting part of the development of modern mathematics. |