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)