Download Neutral geometry

Neutral geometry
And then we add Euclid’s parallel postulate
Saccheri’s dilemma
 Options are:
 Summit angles are right
Wants
 Summit angles are obtuse
 Summit angles are acute
Was able to rule out, and we’ll see how
The hypothesis of the acute angle is
absolutely false, because it is repugnant
to the nature of the straight line!
Rule out obtuse angles:
 If we knew that a quadrilateral can’t have the sum of interior
angles bigger than 360, we’d be fine.
 We’d know that if we knew that a triangle can’t have the sum
of interior angles bigger than 180
180.
 Hold on! Isn
Isn’tt the sum of the interior angles of a triangle
EXACTLY180?
Theorem: Angle sum of any triangle is
l
less
than
th or equall tto 180º
 Suppose there is a triangle with angle sum greater than 180º,
say angle
l sum off ABC is
i 180º + p, where
h p>0.
0
 Goal: Construct a triangle that has the same angle sum, but
one of its angles is smaller than p.
 Why is that enough?
 We would have that the remaining two angles add up to
more than 180º: can that happen?
 Show that any two angles in a triangle add up to less than
180 .
180º
What do we know if we don’t have
P ll l P
Parallel
Postulate???
t l t ???
 Alternate Interior Angle Theorem: If two lines cut
by a transversal have a pair of congruent alternate
interior angles, then the two lines are parallel.
Converse of AIA
 Converse of AIA theorem: If two lines are parallel then the
alternate
l
iinterior
i angles
l cut b
by a transversall are congruent.
 Converse of AIA  Parallel Postulate
If the converse of AIA holds then the sum of the
interior angles of a triangle is 180
So if the parallel postulate holds then we know that the sum of the
interior angles of a triangle is EXACTLY180. But what if we don’t?
Exterior angle theorem: An exterior angle of a
triangle is greater than either remote interior angle.

Proof: Suppose contrary. Then either:
1
1.
DCB  ABC,
ABC or
2.
 DCB < ABC.
Supply the arguments in each case:
1.
We have
B
A
2.
C
Here
B
D
A
C
D
Show that any two angles in a triangle add up to
l than
less
h 180º
B
A
C
D
Consequences:
 Theorem (longer side): Given two non-congruent sides
in a triangle, the angle opposite the longer side is greater
than the angle opposite the shorter side.
side
 Theorem (larger angle): Given two non-congruent sides
in a triangle,
g the angle
g opposite
pp
the longer
g side is ggreater
than the angle opposite the shorter side.
 Theorem (triangle inequality): The sum of the lengths of
any two sides
id off a triangle
i l iis greater than
h the
h llengthh off
the third side.
Consequences:
 Show SAA: If AC  DF,  A   D, and  B   E,
then ABC  DEF.
In neutral geometry: Angle sum of any
ti
triangle
l iis lless th
than or equall tto 180º
 Suppose there is a triangle with angle sum greater than
180º, say angle sum of ABC is 180º + p, where p>0.
 Goal:
G l Construct
C
a triangle
i l that
h h
has the
h same angle
l sum,
but one of its angles is smaller than p.
 Why is that enough?
 We would have that the remaining two angles add up to
more than 180º: can that happen?
 Show that any two angles in a triangle add up to less
than 180º.
Construct a triangle with angle sum as that of ABC
(180º + p), but one of its angles is at most half of m( A)
Saccheri’s dilemma
 Options are:
 Summit angles are right
 Summit angles are obtuse
 Summit angles are acute: