MAT%20360%20Lecture%209

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MAT 360 Lecture 9

• History of the parallel postulate

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Alternate Interior Angle Theorem If alt. int
angle are congruent then lines are parallel.
Measure of angles and segments theorem
Exterior Angle Theorem Exterior angle is greater
than remote interior
The sum of the measures of two angles of a
triangle is less than 180
Saccheri-Legendre Theorem The sum of the interior
angles of a triangle is at most 180

• Hilbert parallelism axiom
• Euclid V
• Converse to Alt. Int. Angle theo
• Sum of int ang of triangle 180
• etc/

Equivalence of parallel postulates are all
equivalent.
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Some attempts to prove Euclids V

• Proclus Measuring distances.
• Wallis Add postulate Given any triangle ?ABC,
  and a segment DE there exists a triangle ?DEF
  similar ( with cong. angles)to ?ABC
• Saccheri Sacheri quadrilaterals ?ABCD (A, B are
  right, C congruent to D). Try prove If Sach.
  quadrilateral not rectangle, then contradiction
• Clairaut Add Axiom Rectangles exist.
• Legendre Accute angle
• Lambert Quadrilaterals with three right angles
• Bolyai

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• Proclus
• Let l and m be parallel lines.
• Let n be a line that intersects m at P. We want
  to show that n intersects l.
• Let Q be the foot of the perp. to l through P.
• If n PQ , we are done.
• Assume n is not PQ . Then there exists Y in n
  and U in m such ray PY is between the rays PU and
  PQ.
• Let X be the foot of the perp. to m through Y.

• As Y moves away from P, the segment XY becomes
  larger and larger (without bound)
• Eventually, XYgtPQ, then Y and P are on different
  sides of l. So, l intersects n.

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• As Y moves away from P, the segment XY becomes
  larger and larger (without bound)
• Eventually, XYgtPQ, then Y and P are on different
  sides of l. So, l intersects m.
• Let Z be the foot of a perpendicular to l
  through Y. Then
• X, Y and Z are collinear
• XZ and PQ are congruent.
• Then when XYgtPQ, XYgtXZ.
• Thus Z is between X and Y.
• So Y and P are on different sides of l.

• Proclus
• Let l and m be parallel lines.
• Let n be a line that intersects m at P. We want
  to show that n intersects l.
• Let Q be the foot of the perp. to l through P.
• If n PQ , we are done.
• Assume n is not PQ . Then there exists Y in n
  and U in m such ray PY is between the rays PX and
  PQ.
• Let X be the foot of the perp. to m through Y.

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Legendres Theorem

• If for any acute angle ltBAC, and any point D in
  the interior of ltA there exist a line through D
  intersecting both rays AB and AC then
• The sum of the interior angles of a triangle is
  180

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(No Transcript)
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In problems 2 and 3 you need to find a way to
work with the sign, taking in to account that the
perfect solution may not exist.

• 1. Write a script to construct an inscribed
  circle in a triangle ?ABC (that is, the incircle
  of a triangle with vertices A, B and C.) The
  Given information should be only three points,
  the vertices of a triangle. (You need to do some
  research to find out the meaning of inscribed
  circle and what the construction is.)
• 2. Write a script to illustrate Menelaus Theorem
  (see exercise H-5 in page 287 of the textbook).
  The Given information should be three points A,
  B and C, and points D, E, F as described on the
  exercise.
• 3. Write a script to illustrate Cevas Theorem
  (see exercise H-6 in page 288 of the textbook).
  The Given information should be three points A,
  B and C, and points D, E, F as described on the
  exercise
• 4. Choose your favorite geometry theorem and
  write a script that illustrate. Do this in a
  separate page of your document and make sure you
  include the statement of the theorem you are
  illustrating.

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