Transcript §21.1 Parallelism

§21.1 Parallelism
The student will learn about:
Euclidean parallelism,
parallelism in absolute
geometry, and special
quadrilaterals.
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Historical Background
Euclid’s Fifth. If a straight line falling on two
straight lines makes the interior angles on the
same side less than two right angles, the two
straight lines, if produced indefinitely, meet on
that side which are the angles less than the two
right angles.
n
l
A
m
 1 +  2 < 180 then
lines l and m meet on
the A side of the
transversal n.
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Playfair’s Postulate
Given a line l and a point P not on l, there exist
one and only one line m through P parallel to l.
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Equivalent Forms of the Fifth
 Area of a right triangle can be infinitely large.
 Angle sum of a triangle is 180.
 Rectangles exist.
 A circle can pass through three points.
 Parallel lines are equidistant.
 Given an interior point of an angle, a line can be
drawn through the point intersecting both sides of
the angle.
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Parallelism
Many early mathematicians, most notably
Proclus (410-450) felt that Euclid’s postulate was
to complicated (perhaps a theorem) and tried to
replace it.
Euclid himself seemed to have difficulty with it
waiting to use it for 38 theorems.
Remember that mathematicians of the time did
not have axiomatic systems with which to work.
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Parallelism
Gauss was the first to recognize the true nature
of the problem and developed a consistent nonEuclidean geometry. However he did not publish
his work for fear of “screaming dullards” .
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Parallelism
János Bolyai developed a geometry that settled
the problem of parallels. “I have discovered such
magnificent things that I am myself astonished at
them … Out of nothing I have created a strange
new world.
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Parallelism
Nicholai Lobachevski independently developed
an elaborate system of non-Euclidean geometry.
It would take 40 years until mathematicians
recognized the importance of the work of these
three great men..
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Euclidean Parallelism
Definition. Two distinct lines l and m are said
to be parallel, l || m, iff they lie in the same
plane and do not meet.
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Theorem 1: Parallelism in
Absolute Geometry
If two lines in the same plane are cut by a
transversal so that a pair of alternate interior
angles are congruent, the lines are parallel.
Notice that this is a theorem and not an axiom
or postulate.
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Parallelism in Absolute Geometry
Given: l, m and transversal t. and
1≅ 2
Proof by contradiction.
(1) l not parallel to m, meet at R.
Prove: l ‫ ׀׀‬m
(2)  1 is exterior angle
Assumption
Def
(3) m  1 > m  2
Exterior angle inequality
(4) → ←
Given  1 ≅  2
l
t
A
1
m
2
B
R
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Three Cases
There are three cases concerning parallelism.
Given a line l and a point P not on l:
1. There exists no line through P parallel to l.
2. There exists one line through P parallel to l.
3. There exists more than one line through P
parallel to l.
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Quadrilateral Review
Know the terms adjacent/consecutive sides,
opposite sides, adjacent/consecutive angles,
opposite angles, convex quadrilateral.
Two quadrilaterals are congruent if their
corresponding angles and sides are congruent.
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Congruency Review
Two quadrilaterals may be proven congruent by
SASAS
ASASA
SASAA
SASSS
Proof is by using the diagonals to form triangles
and using triangle congruency theorems.
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Saccheri Quadrilateral
Let AB be any line segment, and erect two
perpendiculars on the same sides at the endpoints A and
B. Mark off points C and D on these perpendiculars so
that BC = AD. The resulting quadrilateral is a Saccheri
quadrilateral. Side AB is the base, BC and AD are the
legs, and the side CD is the summit. The angles at C and
D are the summit angles.
D
C
A
B
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Saccheri Quadrilateral
The following properties of a Saccheri quadrilateral can
be easily proven:
a. The summit angles are congruent.
b. The diagonals are congruent.
c. The line joining the midpoints of the base and
the summit is the perpendicular bisector of both the
base and summit.
d. If the summit angles are right angles the
Saccheri quadrilateral is a rectangle.
C
D
A
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B
Theorem 2: The summit angles are congruent.
Given: Facts in drawing
Prove: m  C = m  D
(1)  ABCD ≅  BADC
SASAS
(2) C = D
CPCFE.
D
C
A
B
Lambert Quadrilateral
From the previous slide we know that the line joining
the midpoints of the base and the summit is the
perpendicular bisector of both the base and summit.
This line bisects the Saccheri Quadrilateral into two
Lambert Quadrilaterals each with three right angles.
D
C
A
B
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Saccheri Quadrilateral
Saccheri and Lambert investigated the three
possibilities of the summit angles.
The summit angles are obtuse angles.
The summit angles are right angles.
The summit angles are acute angles.
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Theorem 3: The summit angles are not obtuse.
Prove: m  x ≤ 90
Given: Facts in drawing
(1)  BB’C’C constructed below is a Saccheri quadrilateral
associated with Δ ABC
(2) Sum angles of Δ ABC ≤ 180
Previous Thm.
(3) 2x ≤ 180 and x ≤ 90.
Arithmetic
A
B’
B
C’
x
x
C
l
Theorem 4:
The hypothesis of the obtuse angle is not valid
in absolute geometry.
Summary.
• We learned about Euclidean parallelism.
• We learned about parallelism in Absolute
geometry.
• We learned how to prove quadrilaterals
congruent.
• We learned about the Lambert and Saccheri
quadrilaterals.
• We proved the hypothesis of the obtuse angle.
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Assignment: §21.1