Assignment Sheet 1
Please note: Read carefully for the difference between lines, rays, and segments. Not all notation will appear correctly.
Historically, there have been several different approaches to doing geometry, not all of them axiomatic. In order to be able to do geometry, we need a common set of definitions and axioms. Definitions are important because all results depend on the definition used. It is often possible that more than one definition is acceptable. The same can be said of axioms. There are many possible sets of axioms that result in what is typically referred to as Euclidean geometry. I have chosen one particular set of axioms for this class, similar to the ones used by the mathematician Birkhoff, but this is certainly not the only choice. However, it is important that whenever you do any proofs in this class, you do not rely on results we have not assumed or proven. Therefore, you should carefully read all your proofs to be sure that you state the justification for each step.
As we have discussed, it is impossible to define every term in mathematics. The terms point, line, and plane will be undefined for us. Although we can discuss what we mean by these terms, these are the basic objects of study for us and we cannot define them in terms of other things.
Assumptions and postulates:
We assume the properties of the real numbers, of sets and set operations, and of algebra.
Postulate 1: Given any two different points, there is exactly one line that contains both of them. We often restate this as, “Two points determine a line.”
For the next postulate, we assume that we have picked a system of measurement.
Postulate 2 (Distance Postulate)/ Definition of Distance: To each pair of points there is a unique number. This number is called the distance between the two points. For two points P and Q, the distance between them will be written PQ.
Postulate 3 (Number Line Postulate): The points of a given line can be made to correspond to the real numbers in such a way that:
i. Every point of the line corresponds to exactly one real number
ii. Every real number corresponds to exactly one point on the line
iii. The distance between any two points is the absolute value of the difference between the corresponding numbers
Definitions: A coordinate system is a choice of correspondence between points and numbers as described in Postulate 3. The coordinate of a point is the number assigned via this correspondence.
Postulate 4 (Number Line Placement): Given two points on a line, say P and Q, the coordinate system can be chosen so that P is at 0 and the coordinate of Q has a positive value.
Definition: For three points, P, Q, and R, Q is said to be between P and R if PQ + QR = PR.
Any collection of points is said to be collinear if they all lie on the same line.
The absolute value of a real number x, written |x|, is given by: x when x > 0 or x = 0, and –x when x < 0.
The term space or 3-space will be undefined. Informally, we are using the term to mean three-dimensional space.
Definitions: Objects are said to be coplanar if they lie in the same plane.
A segment is a
set of two points together with all the points between them. For two points P and Q, the segment will be
PQ, and P and Q are called the endpoints of the segment.
The length of this segment is
the distance PQ.
A ray PQ, is the set of points P and Q together with all points R such that either R is between P and Q, or Q is between P and R. P is called the endpoint of the ray.
A point R is the midpoint
of a segment
PQ, if R is between P and Q and PR = QR. R, or any object which intersects PQ
at R, is said to bisect PQ.
Postulate 5.1: Every plane contains at least 3 non-collinear points.
Postulate 5.2: Space contains at least 4 non-coplanar points.
Postulate 6: If a plane contains two given points, it contains the line through the two points.
DUE THURSDAY, JANUARY 27:
These problems will be discussed over a period of several classes, but will be collected next class.
i. A square is a rectangle.
ii. A scalene triangle is a triangle with no two sides having a common length.
iii. An equilateral triangle is isosceles.
iv. A square is a rhombus.
FOR CAREFUL WRITE-UP:
Postulate 7: There is at least one plane containing any three given points. If the points are non-collinear, then there is only one such plane.
Postulate 8: If two different planes intersect, then their intersection is a line.
Definition: A set of points is called convex if, for any two points in the set, every point on the segment joining the points is contained in the set.
Postulate 9 (Plane Separation): Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that each set is convex and, given two points, one in each set, the segment joining the points intersects the line.
Definitions: The two sets described in Postulate 9 are called half-planes, and the line is called the edge of each half-plane. The line is also said to separate the plane into two half-planes. Two points that lie in the same half-plane are said to lie on the same side of the line; if they are in different half-planes, they lie on opposite sides of the line.
Postulate 10 (Space Separation): [You will state this one yourself.]
Definitions: The two sets described in Postulate 10 are called half-spaces, and the plane is called a face of each half-space.
An angle is the union of two rays that have the same endpoint. The two rays are each called the sides of the angle, and their common endpoint is called the vertex of the angle.
For any three points, the union of the segments joining them is called a triangle, the segments are called the sides, and the three points are called the vertices of the triangle.
Postulate 11 (Angle Measurement): To every angle there corresponds a number greater than or equal to 0 and less than or equal to 180.
Definition: The measure of an angle is the number assigned through the correspondence in Postulate 11.
(Angle Construction): For any ray AB, such that the ray lies on the edge
of a half-plane, and for any number r, 0 < r < 180, there is exactly one
ray AC with the same endpoint A and
with C in the half-plane, with m<BAC=r.
For r=0 or 180, C will lie on the line
Postulate 13 (Angle Addition): Angle measure is additive for angles which share a common ray: For angles <BAC and <CAD, m<BAC + m<CAD = m<BAD.
Definitions: If two angles, <BAC and <CAD, share a common ray and B, A, and D are collinear, then the two angles form a linear pair. Two angles are supplementary if the sum of their measures is 180, and each angle is said to be a supplement of the other. If two angles form a linear pair and have the same measure, then each angle is a right angle. Two intersecting sets, each of which is a line, a segment, or a ray, are perpendicular if the angles formed by the intersection are right angles.
Two angles are complementary if the sum of their measures is 90, and each angle is said to be a complement of the other.
An angle with a measure less than 90 is called acute, while an angle with measure greater than 90 is called obtuse.
Two angles are congruent if their measures are equal. Two segments are congruent if they have the same length. Two triangles are congruent if there is a correspondence between the angles and segments of each triangle such that the corresponding angles and segments are congruent.
A ray AC is a bisector of angle <BAD if <BAC @ <CAD and the measures of these congruent angles are each not greater than 90.
A median of a triangle is a segment that has one endpoint at a vertex of the triangle and the other endpoint at the midpoint of the opposite side.
Postulate 14 (Supplement Postulate): If two angles form a linear pair, then they are supplementary.
Postulate 15 (SAS Postulate): If there is a correspondence between two triangles such that there are two sides and the angle included between the sides of the first triangle congruent to the corresponding parts of the second triangle, then the triangles are congruent.
Note: We may later take a different look at congruence using isometries of the plane in which we will consider alternative postulates for congruence.
DUE TUESDAY, FEBRUARY 1:
iv. 3 1/2
FOR CAREFUL WRITE-UP
Definition: In a given plane, the perpendicular bisector of a segment is the line that is perpendicular to the segment and intersects the segment at its midpoint.
DUE TUESDAY, FEBRUARY 8:
FOR CAREFUL WRITE-UP:
Definition: Two lines are parallel if they are coplanar and do not intersect, and are skew if they are not coplanar and do not intersect.
A transversal of two lines in a plane is a third line that intersects the two lines in two different points.
DUE TUESDAY, FEBRUARY 15:
FOR CAREFUL WRITE-UP
Euclidean geometry is named for the Greek mathematician
Euclid, and it is what we have focused on so far in this course. One important feature of Euclidean geometry
is known as
Postulate 16 (Parallel Postulate): Through a given point not on a given line, there is at most one line parallel to the given line. [We already proved that there is at least one such line.]
In this week’s problems, we will look at some alternative choices for Postulate 16.
Amazingly, it is not possible to prove the Parallel Postulate from the other 15 postulates and their consequences. This confounded mathematicians for a very long time. We will see later that in non-Euclidean geometry, it is possible to have the other postulates hold true, but to have more than one parallel through a point not on a given line, or to alter things so that there are no parallels, that is, so that lines always intersect.
DUE TUESDAY, FEBRUARY 22:
FOR CAREFUL WRITE-UP:
The quiz on March 1 will cover all the work up through the problems on this page. It will be open notes, and you may quote from any work we have done on the quiz.
Assignment sheet 2
We have, up to now, done Euclidean geometry without use of coordinates. We will now begin to look at geometry in the plane and in 3 dimensions with coordinates and we will think about congruence in a new way.
Definitions: A function T which is 1-1 and has the plane as both its domain and codomain is called a transformation. A transformation T is an isometry if it preserves distance. Two objects A and B in the plane will be said to be congruent (A is congruent to B) if there is an isometry T with T(A) = B. This will be referred to as the “new definition.”
The three most commonly discussed transformations are: reflection, rotation, and translation. We define them here:
A reflection T
across a line m is a transformation satisfying the condition that, for every
point P in the plane, setting P’ = T(P), we have
letting Q be the point of intersection of the lines, PQ=P’Q. A rotation
T around a point R by an angle e, e in radians, -π < e < π is a
transformation such that, for every point P in the plane, if P’=T(P), then m<P’RP = e and P’R = PR. Notice that in the preceding definition
angles have an orientation. A translation T by a directed segment AB is
a transformation such that, for any point C, if C’=T(C), then C’ completes a
parallelogram C’CAB, unless C’ is on the line AB,
in which case C’C = BA and C’B = CA.
With a new definition of congruence, we want to show that the same objects are congruent as before. To do this, we must show objects which were congruent under the new definition are congruent under the old definition, and that objects congruent under the old definition are congruent under the new definition. Even though we redefined congruence, as a result of 49-54, you may continue to use the old theorems about congruence to prove new results.
DUE TUESDAY, MARCH 1
As a result of the above, we know that objects congruent under the new definition are congruent under the old definition. We are not yet able to show that objects congruent under the old definition are congruent under the new definition. The reason for this is that, given two objects congruent under the old definition, we need to find an isometry that maps the first object onto the second. In order for this to happen, we need to know what maps are isometries.
FOR CAREFUL WRITE-UP:
We will begin to do some work using coordinates in the plane (and in three dimensions). We are going to assume that coordinates (x, y) have been established for the plane, and coordinates (x, y, z) have been established for 3 dimensions.
DUE TUESDAY, MARCH 8
FOR CAREFUL WRITE-UP:
IN-CLASS EXPLORATION FOR MARCH 10: What happens to the plane (or objects in the plane, if you prefer) if you perform two reflections? Put another way, what is the result of performing 2 reflections across lines l1 and l2 in the plane? There are three possible cases for the two lines of reflection: l1 and l2 are identical, they are parallel to each other, or they intersect.
DUE TUESDAY, MARCH 15
FOR CAREFUL WRITE-UP:
NOTE: The first exam covers problems 1-75. It will be open notes.
IN-CLASS EXPLORATION ON MARCH 24: We want to know what happens to the plane if you perform three reflections across 3 not necessarily distinct lines. Based on problem 59, you know how three lines can be arranged. For each of these possibilities, and considering different orders, you should come out with two different possible outcomes for the type of transformation you get: a glide reflection, which is a reflection followed by a translation (the most commonly cited example of a glide reflection is an idealized set of footprints, with two symmetric feet leaving tracks equidistant from a fixed line), or a reflection. Use the two problems below to summarize your answer.
The last case to consider is four reflections. It can be proven that the composition of four reflections is either a translation or a rotation. As a result of all this, it turns out that any number of reflections is equivalent to either a reflection, rotation, translation, or glide reflection. From here, there’s a short leap to prove the Theorem: Every distance-preserving transformation T is a composite of reflections. You do not need to prove the theorem. Thus the new definition of congruence is equivalent to the old definition.