Euclidian Geometry Test 2

Name the exposition considered to be the only one more widely used, edited, or studied than the Elements.

The Bible

Lobachevskian Geometry

- Inside the interior of a circle - Give pic of circle plane -Lobachevskian lines are lines that go through the center of the circle or intersect the circle at right angles in L geometry - Denies uniqueness of Euclid's Parallel Postulate - Hyperbolic Geometry - Represented by the Poincare model - Sum of measure of angles in triangle strictly less than 180

Riemannian Geometry

- Lines create circles - Give pic of sphere - No lines are parallel, all intersect - Denies existence of Euclid's Parallel Postulate - Spherical model

Bernhard Riemann

-German -Originator of the class of non-Euclidian geometry called elliptic geometry. -Triangle sums more than 180. -No such thing as parallel lines. -"Riemann metric" extended differenctial geometry to n dimensions, or "Riemannian space."

Euclid

-Greek -Known as the "Father of geometry." -Author of the elements. This book was a comprehensive compilation/explanation of all the known mathematics of his time. -It contains theorems and proofs described in clear, logical, and elegant style, and using only a compass and straight edge. -Euclidian geometry is still as valid today as it was 2,000 years ago. -His "Elements" set the model for mathematical argument, following logical deductions from initial assumptions (axioms and postulates) in order to establish proven theorems. -His problems with parallel postulate led to later non-euclidian geometry. -1st great landmark in history of mathematical thought and organization.

Saccheri

-Italian -Published Euclid Freed of Every Flaw, which was an investigation of Euclid's Parallel Postulate. -Wanted to vindicate Euclid - free him of all error, by proving the 5th postulate. He thought he proved it (but didn't). In the process he came up with some of the first theorems of non-Euclidean geometry, whether he realized it or not. -Created the Saccheri quadrilateral. -Credited with the discovery of non-euclidean geometry.

Nicolai Lobachevsky

-Russian -Developed a geometry in which the 5th postulate was not true, hyperbolic geometry. -This geometry claimed there was more than one parallel through a point to each line (contradicted uniqueness of 5th postulate). -He looked at geometry on curved surfaces. -On such surfaces, triangles have sums less than 180.

What is Absolute Plane Geometry? Give a theorem from this geometry

-The intersection of Lobachevskian (hyperbolic) geometry and Euclidean geometry. -The theorems that are true in both spaces. (theorems will not hold for R) -A geometry independent of the whole question of the parallel postulate -Common ground between Euclidean and Hyperbolic geometry Theorem 1: If 2 lines lie in the same plane and are perpendicular to the same line, then they are parallel. Theorem: sum of angles of triangles is less than or equal to 180 (combination of Euclidean triangle sum and L triangle sum)

Perpendicular

... and the straight line standing on the other is called perpendicular to that on which it stands.

A line is perpendicular to a plane...

...then every plane that contains the given line is perpendicular to the given plane. or: ...if it is perpendicular to every line in that plane.

Euclid's First 5 Postulates

1) A straight line can be drawn from any point to any point 2) A finite straight line can be produced continuously in a straight line 3) A circle may be described with any center and distance 4) All right angles are equal to one another 5) If a straight line falling on two straight lines make the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than 2 right angles.

3 Propositions

1) In any triangle, the greater side subtends the greater angle 2) In any triangle, the greater angle is subtended by the greater side 3) If two straight lines cut one another, they make vertical angles to one another.

Statements Equivalent to 5th Postulate

1) There exists a pair of similar non-congruent triangles. 2) If in a quadrilateral 3 angles are right angles, the 4th is also a right angle. 3) A circle can be passed through any three collinear points. 4) There is no upper limit to the area of a triangle 5) The sum of the angles of a triangle is exactly 180

Axioms

1) Things which are equal to the same thing are also equal to each other. (=) 2) If equals be added to equals, the wholes are equal (+) 3) If equals be subtracted from equals, the remainders are equals (-) 4) Things which coincide with one another are equal to one another 5) The whole is greater than the part (W>P)

The Poincare model for hyperbolic geometry satisfies all the postulates of Euclidean geometry, with the sole exception of which postulate?

5th - The Euclidean Parallel Postulate

The elements consisted of ____ number of books and contains a total of ____ propositions.

13 and 465

Transversal

A line that passes through 2 lines in the same plane at 2 distinct points. (help establish if parallel or not)

How did Euclid (and the Greeks) define postulate?

A postulate is an initial assumption pertaining to the subject at hand.

∆ABC is similar to ∆DEF

AB, AC, BC ∼ DE, DF, EF

Axiom (Early Greeks)

An initial assumption common to all studies

Postulate (Early Greeks)

An initial assumption pertaining to the subject at hand.

What is meant by the axiomatic method in mathematics?

Axiomatic method means a systemic listing of all of the "axioms" that will be used in building the mathematical structure in question.

What is given in the structure of the synthetic approach to geometry that is not given in the metric approach?

Betweenness and congruence

Prove: The diagonals of a Sacherri quadrilateral are congruent.

By SAS, we have ∆BAD≅∆CDA. Therefore segment BD≅AC.

Why is the 5th postulate of Euclid so important?

Because it defined the term "Euclidean Geometry" and because interest in it, lead to so many additional discoveries in geometry as well as in other areas of mathematics.

Restate: Any exterior angle of a triangle is greater than each of its remote interior angles.

Given ∆ABC. If A-C-D, then ∠BCD > ∠B.

Restate: In any triangle, the product and the base of the corresponding altitude is independent of the choice of the base.

Given ∆ABC. Let segment AD be the altitude from A to line BC, and let segment BE be the altitude from B to line AC. Then, AD x BC = BE x AC.

"The Elements"

Euclid's book on geometry and theory of numbers. For over 2,000 years students learned geometry from this book. It served as a model of logical reasoning for everybody. Some of it may have been based on earlier books, many ideas supposedly due to Eudoxus. It was the first book to present geometry in an organized, logical fashion, starting with a few basic assumptions and building on them by logical reasoning.

T/F: The only copy of the Elements found dating back to the author's time was prepared by Theon of Alexandria.

F

T/F: Euclid defines a point as that which has no breadth.

F (a line has no breadth, a point has no part).

T/F: Every Saccheri quadrilateral is a rectangle.

F (only T in Euclidean Geometry)

T/F: The acute angles are equal to one another.

F (right angles)

T/F: The part is greater than the whole.

F (the whole is greater than the part)

How did Euclid (and the Greeks) define axiom?

For the Greek, an axiom was an initial assumption common to all studies.

What does the term "Euclidean Geometry" mean? Are there other types of geometries? Give a brief description of them and clearly explain how they differ from Euclidean geometry.

Geometry that satisfies axioms of Euclidian Geometry, in particular the Parallel Postulate or an equivalent to it. Lobachevskian Riemannian

Who is the author of Euclid Freed of Every Flaw that today would be credited with the discovery of non-euclidean geometry?

Girolamo Saccheri

The Triangle Theorem

Given 3 positive numbers a, b, c. If each of these numbers is less than the sum of the other two, then there is a triangle whose sides have lengths a, b, c.

The SAA Theorem

Given a correspondence between 2 triangles, if 2 angles and a side of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.

SAS Postulate

Given a correspondence between two triangles (or a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.

AAA Similarity Theorem

Given a correspondence between two triangles. If corresponding angles are congruent, then the correspondence is a similarity.

AA Similarity Theorem

Given a correspondence between two triangles. If two pairs of corresponding angles are congruent, then the correspondence is a similarity.

The SAS Similarity Theorem

Given a correspondence between two triangles. If two pairs of corresponding sides are proportional, and the included angles are congruent, then the correspondence is a similarity.

Euclidean Parallel Postulate

Given a line L and an point P not on L, there is one and only one line L' which contains P and is parallel to L. (This is what makes Euclidean Geometry Euclidean Geometry)

The Lobachevskian Parallel Postulate

Given a line L and point P not on L, there are at least two lines L' and L" which contain P and are parallel to L.

Projection of L onto L' in the direction of T (Parallel)

Given two lines L and L' and a transversal T. Let T intersect L and L' in points Q and Q' respectively. Let f(Q) be Q'. For every other point P of L, Let Tp be the line through P, parallel to T. 1) If Tp were parallel to L', then it would follow that T‖L', which is false because T is a transversal to L and L'. Therefore Tp intersects L' in at least one point P'. 2) If Tp=L', if follows that L'‖T, which is false. Therefore Tp intersects L' in at most one point P'. For each point P of L, let f(P) be the unique point P' in which Tp intersects L'. This defines the function f: L→L'. The function f is called the projection of L onto L' in the direction of T.

Vertical projections of L to L' (Perpendicular)

Given two lines L and L' in the same plane, we can define the vertical projection of L to L'. The function is from P to L'. f: L→L' under which to each point P of L there corresponds the foot P'=f(P) of the perpendicular from P to L.

Describe the following construction: Bisecting a line segment AB.

Given two points A and B. First draw a circle C1 with center A, containing B. Then draw C2 with center B containing A. Let AB=a. Since a<2a, it follows that each of the numbers a, a, a is less than the sum of the other two. Therefore the hypothesis of the two-circle theorem is satisfied. Thus, C1 and C2 intersect in two points R and S, lying on opposite sides of segment AB. Therefore, segment RS intersects AB in a point T. The point T is the bisector.

SSS Similarity Theorem

Given two triangles and a correspondence between them. If corresponding sides are proportional, then the corresponding angles are congruent and the correspondence is a similarity.

Restatement of SAA

Given ∆ABC and ∆DEF, ABC↔DEF, if segment AB≅DE, ∠A≅∠D, and ∠C≅∠F, then ∆ABC≅∆DEF

Restatement of Hinge Theorem

Given ∆ABC and ∆DEF, if segment AB≅DE, and segment AC≅DF and m∠A >m∠D, then BC>EF

Restatement of the HL Theorem

Given ∆ABC with right angle at C, and ∆A'B'C' with right angle at C'. If a=a' and c=c', then ∆ABC ≅ ∆A'B'C'

Restatement of AAA

Given ∆ABC, ∆DEF, and a correspondence ABC↔DEF. If ∠A≅∠D, ∠B≅∠E, and ∠C≅∠F, then ∆ABC ∼ ∆DEF.

Restatement of SSS

Given ∆ABC, ∆DEF, and correspondence ABC↔DEF. If a, b, c ∼ d, e, f, then ∆ABC∼∆DEF

Restatement of SAS

Given ∆ABC, ∆DEF, and correspondence ABC↔DEF. If ∠A≅∠D and b, c ∼ e, f then ∆ABC ∼ ∆DEF

When describing The Elements, what does the author mean by the phrase: "Its chief merit lies precisely in the consummate skill with which the proposition were selected and arranged in a logical sequence presumably following from a small handful of initial assumptions."

He meant that the text was well written and organized, establishing a pattern which is used even today.

Name one of the "Imperfections of the Elements" mentioned in the readings

He used but did not mention the concept of betweenness

The Hinge Theorem

If 2 sides of one triangle are congruent to two sides of a second triangle respectively, and the included angles of the first triangle is larger than the included angle of the second triangle, then the opposite side of the first triangle is larger than the opposite side of the second triangle.

Polygonal Inequality

If A1, A2, ...An are any points (n>1), then A1+A2+...+An-1+An ≥ A1An

Alternate Interior Angles

If T is a transversal to L1 and L2, intersecting L1 and L2 in P and Q. And A and D are points of L1 and L2 lying on opposite sides of T, then ∠APQ and ∠PQD are alternating interior angles. (equal if lines are parallel)

Euclid's 5th Postulate

If a straight line falling on two straight lines make the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than 2 right angles.

The modern editions of Euclid's Elements are based on revisions prepared by whom?

Tehon of Alexandria

Point

That which has no part.

"Non Euclidean Geometry"

If it does not have something equivalent to Euclid's 5th postulate. Lobachevskian or Riemannian.

Hypotenuse-Leg Theorem

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another, then the triangles are congruent.

Corresponding Angles

If∠x and ∠y are alternate interior angles, and ∠y and ∠z are vertical angles, then ∠x and ∠z are corresponding angles. (equal if lines are parallel)

The Pythagorean Theorem

In any right triangle, the square of the length of the hypotenuse is the sum of the squares of the lengths of the other two sides.

How and when was Euclid really vindicated? That is, how and when was the question about his fifth postulate finally settled?

In the 19th century when it was shown that his 5th postulate was consistent with and independent of his other postulates. This was done by showing that alternatives to his 5th postulate led to the discovery of other "geometries."

Line

Length without breadth.

Restate: The shortest segment joining a point to a line is the perpendicular segment.

Let L be a line, P a point not on L, Q be the foot of the perpendicular to L through P, and let R be any other point of L. Then segment PQ < PR.

Restate: The perpendicular to a given line, through a given external point is unique.

Let L be a line, and let P be a point not on L. Then there's only one line through P perpendicular to L.

The Basic Similarity Theorem

Let L1, L2, and L3 be 3 parallel lines with common transversals T and T' intersecting them in points A, B, C and A', B', C'. If A-B-C and A'-B'-C' then (BC)/(AB) = (B'C')/(A'B').

Restate: Parallel projections preserve betweenness.

Let f: L↔L' be a parallel projection. If P-Q-R on L, then P'-Q'-R' on L'. Hence, P'=f(P), Q'=f(Q), and R'=f(R).

Restatement of Pythagorean Theorem

Let ∆ABC be a right triangle, with its right angle at C. Then a2+b2=c2.

Surface

That which only has length and breadth.

Riemannian Parallel Postulate

No two lines in the same plane are ever parallel.

Saccheri Quadrilateral

Quadrilateral ABCD is a Saccheri Quadrilateral if ∠A and ∠D are right angles, B and C are on the same side of segment AD, and AB = CD.

Name the two types of non-euclidean geometry discussed in class.

Riemannian and Lobachevskian

Parallel Straight Lines

Straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Differentiate between "synthetic" and "metric" geometry.

Synthetic: congruence and betweenness in structure Metric: Distance and angular measure in structure

T/F: A copy of the Elements was found in the Vatican library

T

T/F: For angles, congruence is an equivalence relation.

T

T/F: Much of the work in the Elements was performed by mathematicians other than Euclid.

T

T/F: Much of the work in traditional high school texts in plane geometry still uses material from the Elements.

T

T/F: Much of the work in traditional high school texts in plane geometry still uses material from the elements.

T

T/F: The extremities of a line are points.

T

Name another alternative to the 5th postulate

There is no upper limit to the area of a triangle.

What is currently used in equivalent to Euclid's fifth postulate?

Through a given point not lying on a line there can be drawn only one line parallel to the given line.

What are the two main instruments used in "constructions" in geometry?

Unmarked ruler and protractor

Right Angle

When a straight line erected on a straight line makes the adjacent angles equal to one another, each angle is called a right angle.

Finish the following statement: If in a quadrilateral ABCD, angles A and B are right angles and sides AD and BC are equal, then...

angles D and C are equal.

The first printing of the Elements was in: a) 1100 b) 1259 c) 2004 d) None of these

d) None of these (1482)

What is given in the structure of the metric approach that is not given in the synthetic approach?

distance and measure for angles (functions)

Which of the following was found in the Elements? a) Results from the number theory b) The Pythagorean Theorem c) Propositions relating to the law of cosines d) None of the above e) All of the above

e) All of the above a) Results from the number theory b) The Pythagorean Theorem c) Propositions relating to the law of cosines

Which one of the following mathematicians are not mentioned anywhere in the article on the Elements? a) Richard Dedekind b) Eudoxus c) Theaetetus d) Theon e) Cole

e) Cole

Which of the following describes the term "absolute geometry"? a) A geometry independent of the whole question of the parallel postulate b) Common ground between Euclidean and Hyperbolic (L) geometry c) Common ground between Euclidean and Elliptic (R) geometry d) Common ground between Elliptic (R) and Hyperbolic (L) geometry e) a and b f) b, c, and d

e) a and b a) A geometry independent of the whole question of the parallel postulate b) Common ground between Euclidean and Hyperbolic geometry

Riemannian geometry denies what property of parallels?

existence - if L is any line and P any point not on L, then there are no lines through P parallel to L.

Which of the following is given as the result for "The Elements" being one of the foremost works ever compiled? a) Its organized mathematical thought. b) The power of the theorems presented. c) Its prototype of the modern mathematical method. d) The few number of imperfections. e) a, b, c f) a, c, d g) all the above

g) all of the above a) Its organized mathematical thought. b) The power of the theorems presented. c) Its prototype of the modern mathematical method. d) The few number of imperfections.

In the modern mathematics of today, what do we consider the difference between an axiom and postulate?

synonymous (Different for Euclid).

Lobachevskian geometry denies what property of parallels?

uniqueness - if L is any line and P any point not on L, then there are at least two lines through P parallel to L.

Restate: Given two lines and a transversal. If a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent.

∠1 ≅ ∠5 implies ∠3 ≅ ∠6. (draw transversal pic where this is true)