Download MAT 3271: Definitions First, remember that there are five

MAT 3271: Definitions
First, remember that there are five accepted undefined terms: “point”, “line”, “between”, “congruent
(to)” (which actually has two usages, one for segments and one for angles), and “incident with” (or its
synonyms, “lie on”, “pass through”, etc.). We can’t go defining things in terms of other things in an
endless regression; we have to start somewhere. The undefined terms may be used in any definition.
1. Given two distinct points A and B, segment AB is the set whose elements are the points A and
←−→
B and all the points lying on line AB that are between A and B. [Note how the undefined term
←−→
“between” must be used. The fact that the points of the segment AB lie on the line AB can be left
←−→
out, as we will have an axiom stating that points between A and B lie on line AB .]
2. Given two points O and A, the circle with center O and radius OA is the set of points, P , with
the property that OP is congruent to OA. [Note that the undefined term “congruent” must be used.
Note also that “segment” must be used, so it had to be defined first. By the transitive property of
congruence, OP is a radius of the circle for every point P that is an element of the circle.]
−−→
3. Ray AB is the union of segment AB and the set of points P such that B is between A and P .
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We say that ray AB emanates from vertex A and is part of line AB.
4. Two rays are opposite if they are distinct, emanate from the same point, and are part of the same
line.
5. An angle with vertex A is a set whose elements are the point A and two non-opposite rays
emanating from A. These rays are called the sides of the angle. [This definition is just a formal way of
collecting in a set the information you need to specify an angle. You don’t really need to include point
A separately, but of course you must specify that the two rays emanate from the same point. Since A
plays a special role as the vertex of the angle, it is customary to include it in the set that defines the
angle. Note that “ray” must be defined before “angle”.]
6. If two angles have a common side and the other two sides are opposite rays, then the angles are
supplementary (or supplements of each other ). [It is not necessary to include labels for the angles and
sides, as the text does, although to do so can aid clarity. I just wanted to make these definitions as
short as possible.]
7. An angle is a right angle if it has a supplementary angle to which it is congruent.
Here are some other words that have been informally defined in the text: Two distinct lines intersect
if there is a point incident with both lines. Two lines are parallel if they do not intersect. A line t is a
transversal to two distinct lines l and m if it intersects both of them.
8. The midpoint of segment AB is the element M of AB with the property that AM is congruent
to BM . [To justify the use of the word “the” in this definition, we will need to first prove that every
segment has a unique midpoint, which we will do.]
9. Two lines l and m are perpendicular if they intersect at a point P and if there is a right angle
with vertex P , each of whose sides is part of l or m.
←−→
10 Line l is the perpendicular bisector of segment AB if l is perpendicular to line AB and passes
through the midpoint of AB. [Clearly it is necessary to define the “midpoint” of AB first. Why don’t
we define the “bisector” of AB, unmodified by the adjective “perpendicular”?]
−−→
11. Given a point D between points A and C, ray AD bisects angle ∠ABC if angle ∠ABD is
congruent to ∠CBD.
12. The points of a given set are collinear if there is a line incident with all of them.
13. The lines of a given set are concurrent of there is a point incident with all of them.
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For the definition of a triangle and related definitions, as well as the definition of vertical angles,
please see the solutions to problem set one.
14. Given four distinct points A, B, C and D, no three of which are collinear, and such that any pair
of the segments AB, BC, CD, and DA have at most an endpoint in common, Quadrilateral ABCD is
the set whose elements are the points A, B, C, and D, and the segments AB, BC, CD, and DA. [Note
the similarity to the definitions of “angle” and “triangle”. Some pictures should convince you that the
conditions on the four points are necessary!]
15. The vertices of quadrilateral ABCD are the points A, B, C, and D.
16. The sides of quadrilateral ABCD are the segments AB, BC, CD, and DA. Two sides are
adjacent if they have a vertex in common. Two sides are opposite if they are not adjacent. A diagonal
of the quadrilateral is a segment whose endpoints are vertices of the quadrilateral, but which is not a
side. [You can also define these terms by just listing the diagonals and the pairs of opposite and adjacent
sides.]
←−→
←−→
←−→
17. Quadrilateral ABCD is a parallelogram if line AB is parallel to line CD and line BC
←−→
is parallel to line DA . [For short, we often just say that a parallelogram is a quadrilateral whose
opposite sides are parallel, but strictly speaking this is not correct, because only lines can be parallel,
not segments.]
Study suggestion and sample questions.
Answers to the sample questions will be posted at http://www.ux1.eiu.edu/ cfcid/ at 12 pm, Sunday
October 12.
Make a list of all the terms above and write the definition to each one. If your definition differs from
the one given in the text or on this sheet, make sure it is equivalent, logically clear, and does not use
any unaccepted terms. If you are not sure, feel free to call me and ask!
Sample questions:
1. Write the definition of midpoint. First, write the definition of every term needed to define midpoint.
If these definitions require that additional terms be defined, define them first, and so on. Write your
final answer in the order in which the terms must be defined.
2. Same question for angle.
3. Same question for right angle.
4. Write the negation, the converse, and the contrapositive of each of the following statements: if
two angles are right angles, then they are congruent; every two distinct lines intersect.
5. Prove from the incidence axioms that there exist three lines that are not concurrent.
6. Prove that, given a line l and a point P not on l, there are two distinct lines through P that
intersect l. [Hint: You will need Incidence Axiom II.]
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