Starred Definitions
parallel lines, and the 4 C's
lines such that no point lies on both,
and
Correct, Clear, Concise, Compelling
Name two civilizations (discussed in class) whose geometry predated the Greeks. How did their geometry differ from Greek geometry?
Answers might include the
Egyptians,
Babylonians,
Chinese,
whose mathematics was more practical and much less justified than Greek mathematics (except perhaps for some Chinese writings).
How an RAA proof of a statement "P" goes, in general
Assume that the negation of P is true, and then use logic to deduce a contradiction of some kind; this forces us to conclude that P was true.
What are the five groups of axioms that Hilbert established
Axioms of
incidence (3),
betweenness (4),
congruence (6),
continuity (eg., Archimedes, Circle-Circle, and/or Dedekind),
parallelism
State the Alternate Interior Angle Theorem.
If a transversal t cuts lines m and n in such a way that two alternate interior angles are congruent, then m and n are parallel.
angle
a pair of (non-opposite) rays emanating from the same point.
Summarize the contributions to geometry of two of the following:
Thales of Miletus
Pythagoras
Plato
Euclid of Alexandria.
Thales: first "mathematician", first on record as demanding proofs.
Pythagoras: founded the Pythagoreans, who knew about irrational numbers (and the Pythagorean Theorem)
Plato: established the Academy and brought together mathematicians; emphasized ideals like circles and lines
Euclid: collected and improved existing geometry/mathematics and published in an axiomatically organized way in the 13 books of Elements; set an example for all who followed.
The converse of "if ABCD is a parallelogram, then ABCD is a quadrilateral"
If ABCD is a quadrilateral, then ABCD is a parallelogram.
The triangle congruence criterion that is listed as a congruence axiom
Side-angle-side (SAS)
State BOTH the Exterior Angle Theorem and the Triangle Inequality.
The Exterior Angle Theorem: In a Hilbert plane, an exterior angle of a given triangle is always greater than either remote interior angles.
The Triangle Inequality: Given any triangle in a Hilbert plane, the sum of the lengths of any two sides of the triangle is greater than the length of the third side.
interior of angle ABC
the set of all points that are both
* on the same side of line AB as C, and
* on the same side of line CB as A.
State Euclid's five postulates (including the uniqueness in the first postulate and giving Playfair's version of the fifth postulate, rather than Euclid's original).
I: Between any two points there is a unique line.
II: Each line can be extended indefinitely in either direction.
III: It is possible to draw a circle with any radius at any center.
IV: All right angles are "equal".
V (Playfair's axiom): Given any line m and any point P not on m, there exists exactly one line through P that is parallel to m.
The contrapositive of "If line m is tangent to the circle C, then m intersects C in exactly one point."
If line m intersects the circle C in fewer than or more than one point, then m is not tangent to C.
Possible answers come from:
SSS
ASA
SAA
Hypotenuse-leg (for right triangles)
Give at least three statements equivalent to, but not the same as, Euclid's Postulate V.
(See the results near the end of Chapter 4.)
AB < CD
there exists a point E such that C*E*D and AB is congruent to CE.
For at least two of the following Alexandrian mathematicians, name a mathematical contribution :
Archimedes
Eratosthenes
Apollonius of Perga
Heron of Alexandria.
Archimedes: approximating pi using polygons inscribed in/circumscribed about a circle; study of cylinder and the sphere.
Eratosthenes: computing the size of the earth.
Appolonius: "The Conics"
Heron: Area of a triangle; A = (1/4)sqrt[s(s-a)(s-b)(s-c)]
What is the hyperbolic parallel property?
State two theorems about isosceles triangles.
If a triangle is isosceles (i.e., if a pair of sides is congruent), then the base angles of those sides are also congruent.
If two angles are congruent in a triangle, then the sides opposite these angles are also congruent (i.e., the triangle is isosceles).
Describe the Saccheri-Legendre Theorem.
In any Archimedean Hilbert plane, the sum of the degree measures of the three angles in any triangle is less than or equal to 180 degrees.
Hilbert plane
a model of the incidence, betweenness, and congruence axioms.
List the four famous, anciently-unsolved construction problems of antiquity, and describe the modern outcome of these problems.
* Trisecting any angle,
* squaring any circle,
* duplicating any cube, and
* constructing any n-gon
(each of these using only a compass and straightedge);
all were shown to be impossible--the 1800's proofs relied on abstract algebra and number theory.
Describe how
formal systems (consisting of undefi?ned terms, axioms, and theorems),
interpretations, and
models are related to one another.
Given a formal system,
an interpretation is an assignment of meaning to the undefined terms, and
a model of the formal system is an interpretation that satisfies all the axioms (so, it's a "world" in which the theorems hold true).
Explain precisely (using only undefined terms and/or axioms) what it means for two points A,B to be on the same side of a line m.
A,B are on the same side of m if the segment AB contains no point in common with m; in other words, neither A nor B nor any point between them belongs to m.
What is "neutral" geometry, and what is one reason it's worth studying?
The collection of theorems that can be proved without making any assumptions as to the validity of any particular parallel property (Euclidean or otherwise).
One benefit: If the Euclidean parallel property turns out to be a theorem of Euclidean geometry, then we will have proof that the ancient critics of Euclid's Postulate V were right.
Another benefit: Euclidean geometry is just one example of a Hilbert plane. Any theorems proved without assuming anything about a parallel property will be true in other, possibly non-Euclidean Hilbert planes as well.