Transcript MAT360 Lecture 9

MAT 360 Lecture 9
History of the parallel postulate
Alternate Interior Angle Theorem
If alt. int angle are congruent then lines are parallel.
Exterior Angle Theorem
Exterior angle is greater than remote interior
Measure of angles and
segments theorem
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
Equivalence of parallel
postulates:
are all equivalent.
Hilbert parallelism axiom
Euclid V
Converse to Alt. Int. Angle theo
Sum of int ang of triangle 180
etc/
Some attempts to prove
Euclid’s V
• Proclus: Measuring distances.
• Wallis: Add postulate: “Given any triangle ΔABC,
and a segment DE there exists a triangle ΔDEF
similar (= with cong. angles)to ΔABC”
• 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
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, XY>PQ, then Y
and P are on different sides
of l. So, l intersects n.
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.
•As Y moves away from P,
the segment XY becomes
larger and larger (without
bound)
•Eventually, XY>PQ, 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 XY>PQ, XY>XZ.
Thus Z is between X and Y.
So Y and P are on different
sides of l.
Legendre’s Theorem
If for any acute angle <BAC, and any point
D in the interior of <A 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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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
ΔABC (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 H5 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 Ceva’s 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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