Math 467 T/F
It is an immediate consequence of Axiom C-2 that if AB=CD, then CD=AB.
True
A*B*C is logically equivalent to C*B*A.
False
An implication is logically equivalent to its converse.
False
Archimedes was the first to develop a theory of proportions valid for irrational lengths.
False
By definition, a "right angle" is a 90 degree 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.
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
It was unnecessary for Euclid to assume the parallel postulate because the French mathematician Legendre proved it.
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 meaning of the Greek word "geometry" is "the art of reasoning well from badly drawn diagrams".
False
The precise technology of measurement available to us today confirms the Pythagoreans' claim that √2 is irrational.
False
We can use Pappus' method to prove the converse of the theorem on base angles of an isosceles triangle if we first prove the angle-side-angle (ASA) criterion for congruence.
False
angle ABC is a subset of triangle ABC
False
segment AB = ray AB union ray BA
False
If a space S has 6 different points, then it must have exactly 16 different lines.
False ((6*5)/2 = 15)
"Axioms" or "postulates" are statements that are assumed, without further justification, whereas "theorems" or "propositions" are proved using the axioms.
True
A "transversal" to two lines is another line that intersects both of them in distinct points.
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
Although pi is a Greek letter, in Euclid's Elements it did not denote the number we understand it to denote today.
True
An "angle" is defined as the space between two rays that emanate from a common point.
True
Descartes brought algebra into the study of geometry and showed he could solve every geometric problem with his method.
True
Euclid attempted unsuccessfully to prove the side-angle-side (SAS) criterion for congruence by a method called "superposition".
True
Four point geometries fail to satisfy the betweenness axioms.
True
If A*B*C, then ray BC is a subset of ray AB.
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 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
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 ancient Greek astronomers did not believe that three-dimensional euclidean geometry was a 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 implication "If 4 is odd, then 4 > 7" is true.
True
We call √2 an "irrational number" because it cannot be expressed as a quotient of two whole numbers.
True