1 §21.1 Parallelism The student will learn about: Euclidean parallelism, and special...

1

§21.1 Parallelism

The student will learn about:

Euclidean parallelism,

and special quadrilaterals.

parallelism in absolute geometry,

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Historical BackgroundEuclid’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.

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A

ln

m

1

2

1 + 2 < 180 then lines l and m meet on the A side of the transversal n.

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Playfair’s Postulate

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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.

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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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ParallelismMany early mathematicians, most notably Proclus (410-450) felt that Euclid’s postulate was to complicated (perhaps a theorem) and tried to replace it.

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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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ParallelismGauss was the first to recognize the true nature of the problem and developed a consistent non-Euclidean geometry. However he did not publish his work for fear of “screaming dullards” .

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ParallelismJános 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

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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

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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 GeometryGiven: l, m and transversal t. and

1 ≅ 2 Prove: l ׀׀ m

BR

l

m

1

tA

2

Proof by contradiction.

(1) l not parallel to m, meet at R. Assumption

(2) 1 is exterior angle Def

(3) m 1 > m 2 Exterior angle inequality

(4) → ← Given 1 ≅ 2

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Three Cases

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There are three cases concerning parallelism.

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.

Given a line l and a point P not on l:

Quadrilateral Review

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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.

Congruency Review

Two quadrilaterals may be proven congruent by

SASAS

ASASA

SASAA

SASSS

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Proof is by using the diagonals to form triangles and using triangle congruency theorems.

Saccheri Quadrilateral

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Let 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 is the base, are the legs, and the side is the summit. The angles at C and D are the summit angles.

AB

AB BC and AD

CD

A

D C

B

Saccheri Quadrilateral

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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.

A

D C

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.

A

D C

B

Lambert Quadrilateral

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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.

A

D C

B

Saccheri Quadrilateral

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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.

Theorem 3: The summit angles are not obtuse.

Given: Facts in drawing Prove: m x ≤ 90

(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 C

B’ C’

x x

l

Theorem 4:

The hypothesis of the obtuse angle is not valid in absolute geometry.

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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.

Assignment: §21.1