True/False Questions
The Saccheri-Legendre theorem tells us that some triangles exists that have angle sums less than 180º and some triangles exists that have angle sums equal to 180º
Depends
A * B * C is logically equivalent to C * B * A
False
All Euclidean planes are isomorphic to one another
False
An "angle" is defined as the space between two rays that emanate from a common point
False
An exterior angle of a triangle is any angle that is not in the interior of the triangle
False
Archimedes was the first to develop a theory of proportions valid for irrational lengths
False
Archimedes' Axiom is independent of the other 15 Axioms for Euclidean geometry given in this book
False
By definition a "right" angle is a 90º angle
False
By definition, a line m is "parallel" to a line l if for any two points P, Q on m, the perpendicular distance from P to l is the same as the perpendicular distance from Q to l.F
False
Descartes brought algebra into the study of geometry and showed he could solve every geometric problem with his method.
False
Euclid provided constructions for bisecting and trisecting any angle
False
Hilbert's axiom of parallelism is the same as the Euclidean parallel postulate given in Chapter 1.
False
If A, B, and C are distinct collinear points, it is possible that both A*B*C and A*C*B.
False
If ABC is any triangle and if a perpendicular is dropped from C to line AB, then that perpendicular will intersect line AB in a point between A and B.
False
If two triangles have the same angle sum, they are congruent
False
It is a theorem in neutral geometry that if l and m are parallel lines, then alternate interior angles cut out by any transversal to l and m are congruent to each other.
False
It is a theorem in neutral geometry that if l || m, m || n, then l || n
False
It is impossible to prove in neutral geometry that parallel lines exist
False
It was unnecessary for Euclid to assume the parallel postulate because the French mathematician Legendre proved it
False
Legendre proved in neutral geometry. that for. any acute angle A and any point D in the interior of A, there exists a line through D not through A which intersects both sides of A.
False
Most of the results in Euclid's Elements were discovered by Euclid himself.
False
One of the congruence axioms asserts that if congruent segments are "subtracted" from congruent segments, the differences are congruent.
False
One of the congruence axioms is the side-side-side (SSS) criterion for congruence of triangles.
False
The "line separation property" asserts that a line has two sides
False
The ASA criterion for congruence of triangles is one of our axioms for neutral geometry
False
The Saccheri-Legendre theorem is false in Euclidean geometry because in Euclidean geometry the angle sum of any triangle is never less than 180•
False
The alternate interior angle theorem implies, as a special case, that if a transversal is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
False
The alternate interior angle theorem states that if parallel lines are cut by a transversal, then alternate interior angles are congruent to each other.
False
The meaning of the Greek word "geometry" is "the art of reasoning well from badly drawn diagrams."
False
The notion of congruence for two triangles is not defined in this chapter
False
The precise technology of measurement available to us today confirms the Pythagoreans' claim that sqrt(2) is irrational
False
Theorem 4.4 shows that Euclid's fifth postulate is a theorem in neutral geometry.
False
We call sqrt(2) an "irrational number" because it cannot be expressed as a quotient of two whole numbers
False
"Axioms" or "postulates" are statements that are assumed, without further justification, whereas "theorems" or "propositions" are proved using the axioms
True
A "Lambert quadrilateral" is a quadrilateral having at least three right angles
True
A "Saccheri" quadrilateral is a quadrilateral ABDC such that angles CAB and DBA are right angles and AC $\cong$ BD
True
A "transversal" to two lines is another line that intersects both of them in distinct points
True
A Hilbert plane is any model of the incidence, betweenness, and congruence axioms
True
A Lambert quadrilateral can be "doubled" to form a Saccheri quadrilateral, and a Saccheri quadrilateral can be "halved" to form a Lambert quadrilateral
True
A great many of Euclid's propositions can be interpreted as constructions with straightedge and compass, although he never mentions those instruments explicitly.
True
A quadrilateral that is both a Saccheri and a Lambert quadrilateral must be a rectangle
True
AB<CD means that there is a point E between C and D such that AB is congruent to CE
True
Although pi is a Greek letter, in Euclid's Elements it did not denote the number we understand it to denote today.
True
Archimedes' axiom is used to measure segments and angles by real numbers
True
Euclid attempted unsuccessfully to prove the side-angle-side (SAS) criterion for congruence by a method called "superposition".
True
Euclid's fourth postulate is a theorem in neutral geometry.
True
If a Hilbert plane satisfies Aristotle's Axiom, then the fourth angle in a Lambert quadrilateral in that plane cannot be obtuse.
True
If line m is parallel to line l, then all the points on m lie on the same side of l.
True
If points A and B are on opposite sides of a line l, then a point C not on l must be either on the same side of l as A or on the same side of l as B.
True
If we were to take Pasch's theorem as an axiom instead of the separation axiom B-4, then B-4 could be proved as a theorem.
True
In Axiom B-2, it is unnecessary to assume the existence of a point E such that B*D*E because this can be proved from the rest of the axiom and Axiom B-1, by interchanging the roles of B and D and taking E to be A.
True
In sphere interpretation, where "lines" are interpreted to be great circles, Euclid V holds, yet the Euclidean parallel postulate does not.
True
In the statement of Axiom C-4, the variables A, B, C, A', and B' are quantified universally and the variable C' is quantified existentially
True
It is a Theorem in Neutral Geometry that vertical angles are congruent to each other
True
It is a theorem in neutral geometry that every segment has a unique midpoint
True
It is a theorem in neutral geometry that given any point P and any line l, there is at most one line through P perpendicular to l.
True
It is a theorem in neutral geometry that if a rectangle exists, then the angle sum of any triangle is 180º
True
It is an immediate consequence of Axiom C-2 that if AB is congruent to CD then CD is congruent to AB
True
The Euclidean Parallel Postulate states that for every line l and for every point P not lying on l there exists a unique line m through P that is parallel to l
True
The SSS criterion for congruence of triangles is a theorem in neutral geometry
True
The ancient Greek astronomers did not believe that three-dimensional Euclidean geometry was an idealized model of the entire space in which we live because they believed the universe is finite in extent, whereas Euclidean lines can be extended indefinitely.
True
The ancient Greeks were the first to insist on proofs for mathematical statements to make sure they were correct.
True
The gap in Euclid's attempt to prove Theorem 4.2 can be filled using our axioms of betweenness.
True
The notion of one ray being "between" two others is undefined
True
Wallis' postulate implies that there exist two triangles that are similar but not congruent.
True
It is impossible to prove in neutral geometry that rectangles exist
True (sorry Joseph)
