3.5 Segments, Rays, Angles, and Triangles 67
If you review the definitions of AB, AB , LBAC, and AABC, you will see that all of these definitions are based on the idea of betweenness. The proofs of Theorems 1 through 5 must, therefore, be based on Theorems B-1 through B-5.
A word of caution: Here and hereafter, the symbol = is going to be used in one and only one sense; it means "is exactly the same as." Thus, when we write AB = BA, we mean that the sets AB and BA have exactly the same elements.
Finally, a few remarks may be in order about the way in which we have de-fined the idea of an angle. Under our definition, an angle is simply a set which is the union of two noncollinear rays with the same end-point. Angles, in this sense, are quite adequate for the purposes of Euclidean geometry.
Much later, in analytic geometry and in trigonometry, we shall need to talk about directed angles in which the initial side can be distinguished from the ter-minal side, like this:
Figure 3.12
An angle, in this sense, is not a set of points but rather an ordered pair (AB, AC) of rays; thus (AB, AC) is different from (AC, AB). For directed angles, we allow the possibility that the sides are collinear, and also the possibility that the sides are the same. We have not used this more complicated idea of an angle, because at the present stage we have no use for it. For example, the angles of a triangle never consist of two collinear rays, and there is no natural way to assign direc-tions to them.
For the purposes of this book, there are good reasons for ruling out "zero angles" and "straight angles." In the first place, these terms are superfluous: a zero angle is simply a ray, and a straight angle is a line. In the second place, an-gles, rays, and lines are different figures, in important ways; and if we used the same word "angle" to apply to all three, then we would continually be involved in discussions of special cases. (Contrary to popular impression, Euclid did not use "straight angles.")
68 Distance and Congruence
3. Prove Theorem 2.
4. Prove Theorem 3.
*5. Prove the following. Let A and B be two points, and let D, E, and F be three non-collinear points. If AB contains only one of the points D, E, or F, then each of the lines DE, DF, EF intersects AB in at most one point.
*6. Prove the following. If LABC = ADEF, then each of the lines AB , BC, AC contains two of the points D, E, and F.
*7. Show that for any AABC, we have AB 11 LABC = AB. That is, the only points of AB that lie on the triangle are the points of the side AB.
*8. Prove the following. If LABC = ADEF, then each side of AABC contains two of the points D, E, and F.
9. Show that A is not between any two points of /ABC.
10. Prove Theorem 5.
3.6 Congruence of Segments
The intuitive idea of congruence, for any two figures at all, is always the same.
Two figures F and G are congruent if one can be moved so as to coincide with the other. Thus two equilateral triangles of the same size are always congruent;
two circles of the same radius are always congruent; two squares of the same size are always congruent, and so on.
LA 00
Figure 3.13
In the same way, two segments of the same length are always congruent.
. • • •
A B C D
Figure 3.14
Here, by the length of a segment, we mean the distance between its end points.
Our problem, in our mathematical study of congruence, is to formulate the idea in sufficiently exact form to be able to prove things about it. In the present
3.6 Congruence of Segments 69 section, we shall do this for the case in which the figures are segments. Later we shall do the same for the case in which the figures are angles; and still later, we shall discuss triangles. Finally, in the chapter on rigid motion, we shall discuss congruence in a form sufficiently general to apply to any two sets of points.
We start with our official definition.
DEFINITION. Let AB and CD be segments. If AB = CD, then the segments are called congruent, and we write AB = CD.
On the basis of this definition, it is easy to prove the familiar and fairly triv-ial facts about congruence of segments.
A relation —, defined on a set A, is called an equivalence relation if the fol-lowing conditions hold.
(1) Reflexity. a — a, for every a.
(2) Symmetry. If a — b, then b — a.
(3) Transitivity. If a — b and b c, then a — c.
■ THEOREM C-1. For segments, congruence is an equivalence relation.
That is, every segment is congruent to itself; if AB = CD, then CD = AB; if AB = CD and CD == EF, then AB -=-• EF.
Proof?
■ THEOREM C-2. The Segment-Construction Theorem. Given a segment AB and a ray CD. There is exactly one point E of CD such that AB = CE.
A
AB-CE
Figure 3.15
That is, starting at the end point of a ray, you can measure off a segment of any desired length, and the resulting segment is unique.
PROOF. By the ruler placement theorem, set up a coordinate system f for the line CD, in such a way that f(C) = 0 and f(D) > 0.
C D E
0 CD x = AB
Figure 3.16
In the figure, we have indicated that the number CD is the coordinate of the point D, and this is correct, because f(D) > 0. If E is a point of CD, then
70 Distance and Congruence
CE = AB if and only if f (E) = AB as in the figure. Thus CE = AB if and only if E = f -'(AB). There is exactly one such point f -'(AB), and therefore there is ex-actly one such point E.
The following theorem says, in effect, that if congruent segments are laid end to end, the resulting segments are congruent.
Figure 3.17
IN THEOREM C-3. The Segment-Addition Theorem. If (1) A-B-C,
(2) A' -B' -C' , (3) AB = A' B' , and
(4) BC = B'C', then
(5) AC = A'C'.
We also have a converse.
IN THEOREM C-4. The Segment-Subtraction Theorem. If (1) A-B-C, (2) A'-B'-C', (3) AB = A' B' , and (4) AC = A'C', then (5) BC = B'C'.
These theorems can most conveniently be proved by means of the defini-tion of betweenness. You should work out the proofs in full.
Note that we have called Theorem C-4 a converse of Theorem C-3, rather than the converse of Theorem C-3. The reason is that most theorems have more than one converse (each of which, of course, may or may not be true).
For statements of the form P Q, where P and Q are propositions, the situ- ation is simple. The converse of the implication P Q is the implication Q P. A theorem may, however, be stated in the following form: "If (a), (b), and (c), then (d), (e), and (f)." This says that
[(a) and (b) and (c)] [(d) and (e) and (f)].
In this case, any statement that you get by interchanging part of the hypothesis and part of the conclusion is called a converse. Thus, to get Theorem C-4 from Theorem C-3, we moved (5) into the hypothesis, and moved (4) into the con-
3.6 Congruence of Segments 71
clusion. Theorem C-3 has three more converses. You should state them and find out which of them are true.
If A-B-C, and AB =--- BC, then B is a mzdpoznt of AC. The following theorem justifies us in referring to B as the midpoint.
■
THEOREM C-5. Every segment has exactly one midpoint.PROOF. Given AC. By the ruler placement theorem, take a coordinate system f, for the line AC, such that f (A) = 0 and f(C) > 0.
A B
•
0 x AC
Figure 3.18 If B is between A and C, then
AB = Ix — 01 = x and
BC = 1AC — xl = AC — x .
Thus, for the case where A-B-C, the condition AB --=---- BC is equivalent to the condition
x = AC — x , or
2x = AC , or
x= AC
7.
There is exactly one such number x, and therefore there is exactly one such point B. ❑
C
B
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