AN IMPROVEMENT ON SSA CONGRUENCE FOR GEOMETRY AND TRIGONOMETRY

AN IMPROVEMENT ON SSA CONGRUENCE FOR GEOMETRY AND TRIGONOMETRYAuthor(s): SHRAGA YESHURUN and DAVID C. KAYSource: The Mathematics Teacher, Vol. 76, No. 5 (May 1983), pp. 364-367, 347Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963527 .

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AN IMPROVEMENT ON SSA CONGRUENCE FOR GEOMETRY AND TRIGONOMETRY

By SHRAGA YESHURUN Bar-Han University

Ramat-Gan, Israel 52100

and DAVID C. KAY University of Oklahoma

Norman, OK 73019

Many textbooks mention what might be called three and a half basic congruence theorems: SAS, ASA, SSS, and HL (for right triangles). Perhaps because of the am

biguity involved in the SSA condition, this criterion for congruence is consistently ne

glected. A recent article of Litwiller and Duncan (1981) investigated the SSA theo rem, but it seems desirable to extend and, at one point, to correct the result given there. It is frequently said that a good problem in mathematics is one that gener ates new problems. The same is true for

good investigations. The one by Litwiller and Duncan gave rise to three further

points to be explored: (1) an improvement of the SSA congruence theorem in their article that works better in trigonometry, (2) a discussion of the failure of SSA in

spherical geometry, and (3) an extension of SSA to spherical geometry as well as hy perbolic geometry.

An SSA theorem in Euclidean geometry and trigonometry

Whereas the SSA theorem of Litwiller and Duncan dealt with conditions on

angles, we propose to use a segment in

equality to disallow the famous ambiguous case. Our proposition is as follows :

// the first and second sides and the angle opposite the first side of one triangle are

congruent to the first and second sides and the angle opposite the first side of another

triangle, respectively, such that the length of the first side is not less than that of the

second, then the triangles are congruent.

In this form the SSA congruence theo rem is valid in Euclidean geometry and is a

member with "equal rights" in the set of basic congruence theorems for geometry: SAS, ASA or SAA, SSS and SSA. Obvi

ously, HL for congruence of right triangles is a special case of SSA in the formulation

given above. The only difference between SSA and the

other congruence theorems is in the proof. Of course, it depends on which axiomatic

system one takes for the basis of study. Some systems take one of these congruence theorems as an axiom, e.g., postulate 15 in the SMSG postulates (SMSG 1966, p. 187).

Other systems allow the proof of all the

congruence theorems on the basis of the axioms and definition of congruence, e.g., Coxford and Usiskin (1975, pp. 234-57). In

any case, the three basic congruence cri

teria, SAS, ASA, and SSS, and the special case of the fourth, HL, are either axioms or

directly provable. On the contrary, the gen eral SSA theorem presented above involves

inequalities among the sides and angles of

triangles, like some textbook proofs of SAA. This may be the basis for the general exclusion of the SSA congruence theorem from textbooks. (It was recently called to our attention that our theorem does appear, with proof by pictures, in Trig onometry with Tables by Welchons and

Krickenberger [New York: Ginn & Co., I960].) Some authors, however, even deny in express terms its existence (see Coxford and Usiskin [1975]).

Of course it is true that SSA is not a

congruence theorem without adding the extra condition

such that the length of the first side is not less than that of the second.

Whether or not to include the SSA congru ence theorem, together with the mentioned additional condition, among the basic

364 Mathematics Teacher



congruence theorems is, to some extent, a

matter of taste. The authors advocate its

inclusion. As mentioned above, the proof seems not

to be found in English textbooks; so it will

be presented here. Let AB^XY, BC^

YZ, BC > AB, and LA ? LX (as in fig.

1). We assert that AABC s A .

false, and so the indirect proof is complete.

An examination of our proof shows that

the only triangles satisfying the conditions in the congruence part of the SSA hypoth esis but failing to be congruent are precise

ly those that fail our special inequality con

dition. A similar statement could be made

regarding the LitWiller-Duncan SSA theo

rem; so our result, arid theirs, can be re

garded as "best possible" SSA congruence theorems. However, whereas the SSA cri

terion relates to solutions of triangles in

trigonometry, our result covers all triangles with SSA data yielding unique solutions. In

particular, the Litwiller-Duncan theorem

requires the presence of a right or obtuse

angle, whereas ours applies to triangles in

which all angles are acute. To clarify this discussion, we have in

cluded in table 1 the various SSA data for

triangles constituting the ambiguous case

(see Ayres [1954]). Let a be the length of the first side of a triangle, b that of the

second, and a the measure of the angle op

posite the first side. The first two lines in

table 1 represent compliance with our spe cial inequality for SSA; the others involve a

violation of that inequality.

Fig. 1. The indirect proof of SSA congruence theo rem

Proof. If AC ? XZ9 our assertion is

proved. If not, without loss of generality we

can assume that AC > XZ, and so there

exists De AC such that AD^XZ. This

implies that AABD ? AXYZ by SAS and by the given condition BD^YZ^ BC. From here we see that ABCD is isosceles with BD s BC and LC s lD.

However, mLD > m LA (by the exterior

angle theorem), which implies that

mLC > m LA frorh the fact that the base

angles of an isosceles triangle have equal measures. By the theorem "longer side op

posite the larger angle," it follows that AB > BC, although it was given that BC > AB. This contradiction implies that our denial of the congruence AC ? XZ is

TABLE 1 The Number of Real Solutions of the

SSA Construction or Computation Problem

Number of real solutions if the angle opposite Relationships

of size

among the data acute

the first side is?

right or obtuse

a > b a = b

b sin a < a< b a ? b sin a*

a< b sin a

* This condition implies ? =

SSA but an SAA problem. 90?; so this is not really an

A question not generally dealt with is

this:

If we choose a > 0, b > 0, and 0? < a < 180? at random, how often do we have two, one, or no real solutions?

This question has been investigated else

May 1983 365



TABLE 2 Frequency of Two, One, or No

Re?l Solutions for SSA Problems

Experimental probability or relative frequency (one million cases)

Number of Theoretical First Second real solutions probability Computer run computer run

2 0.0908 0.0907 0.09?5 1 0.5000 0.5009 0.5005 0 0.4092 0.4084 0.4090

where (Yeshurun and Merzbach, in press), and we quote in table 2 only the results.

The SSA theorem in spherical geometry

The essence of the Litwiller-Duncan the orem is that SSA exists as a congruence theorem if each member of any pair of cor

responding angles in the given triangles is either a right or an obtuse angle. The au thors assert that "these results would apply equally well in non-Eculidean geometry." A few clarifying remarks are in order. All of us know very well that a thousand exam

ples do not prove a theorem but that a

single counterexample disproves it. A coun

terexample can also pave the way for ex

tensions of theorems that are false in a more restrictive setting. Let us consider one

that applies to the SSA theorem in spheri cal geometry.

Let triangles ABC and XYZ on the

sphere in figure 2 be so arranged that

BC= YZ, which is the "equator "

A is the "north pole," X is the "south pole," and BC > YZ. (This example appears in a

college-level text, now out of print, by Kay (1969). The angles B9 C, 7, and will then be right angles, with AB = XY = one

quarter of the great circle, and AC = XZ as well. Hence the triangles satisfy the hy pothesis of the SSA theorem of Litwiller and Duncan, as well as ours, but the trian

gles are not congruent. In spite of this ex

ample, however, it is not to be concluded that the SSA theorem is false on all parts of the sphere, for we shall subsequently es tablish a theorem that includes a large class of spherical triangles. This difficulty does

distinguish SSA from the three basic cri teria SAS, ASA, and SSS, which all hold in

elliptic geometry without restriction. (We can easily see from the same example that SAA fails in elliptic geometry in general.)

Fig. 2. Counterexample of SSA theorem on a sphere

The example above also points up a common misconception regarding non Euclidean geometry. It is often thought that the only substantial difference among the three geometries?i.e., elliptic or Rie

mannian, parabolic or Euclidean, and hy perbolic or B?lyai-Lobachevskian?is in the axiom of parallels or, under Playfair's presentation, in the number of parallels to a given line through a given point not be

longing to the given line. The number of such parallels in elliptic geometry is zero, in

parabolic, one, and in hyperbolic, more than one. Although this difference is dra

matic and important in its effect, other sub stantial differences show up in areas having no apparent connection with parallelism.

366 Mathematics Teacher



In deciding whether a certain result in Eu clidean geometry belongs to non-Euclidean

geometry, therefore, it is not enough to examine the proof to see if the Euclidean axiom of parallels was used.

The original five postulates of Euclid

(Heath 1956) are not satisfactory for our times. Several other axiom systems have been constructed for Euclidean geometry (see a selected list in Yeshurun [1982]). A

system of twelve axioms for all three geom etries has been proposed by Kay (1969, p. 43). In this system, for example, axiom 4 is

given as follows :

Each pair of distinct points lies on at least one line, and

if their distance is less than the least upper bound of the set of all distances, that line is unique.

This axiom does not pertain to the parallel postulate yet is different for Euclidean and

elliptic geometries?it allows more than one line to contain the same pair of points as elements if the set of distances is bound ed (as it is in spherical geometry). Indeed, it is later proved in Kay (1969) that if the least upper bound for distances is a finite number d, then every pair of points at a distance d apart belongs to infinitely many lines, and the proof does not use the paral lel axiom for spherical geometry in any

way?it is, rather, the result of the particu lar coordinatization axiom assumed for lines and the other Euclideanlike axioms.

SSA in non-Euclidean geometry

It is interesting that much of the theory of triangles developed by Bolyai and Lo bachevski for hyperbolic geometry is valid for a large portion of a sphere. To be spe cific, if one restricts attention to those

triangles with sides of length less than half the least upper bound of the set of all dis tances (the number d mentioned above),

which we may name permissible triangles, then all the theory in common between Eu clidean and hyperbolic geometry remains intact for spherical geometry. Note that in Euclidean or hyperbolic geometry, all

triangles are permissible. The starting point is the (weak) exterior

angle theorem (see Kay [1969, p. 99]):

In any permissible triangle, an exterior angle is greater than either opposite interior angle.

From this proposition can be proved the

following familiar chain of theorems (in fact, the proofs can be essentially lifted from Euclid):

A. If one side of a permissible triangle is

greater than another, the angle opposite the greater side is the greater angle.

B. (Converse of A). If one angle of a per missible triangle is greater than another, the side opposite the greater angle is the

greater side.

C. There exists at most one obtuse or

right angle in any permissible triangle. D. The base angles of a permissible isos

celes triangle are acute.

E. The SAA congruence criterion is valid for permissible triangles.

It is clear that the proof given earlier for the SSA theorem in Euclidean geometry can now be repeated verbatim for per missible triangles in non-Euclidean geome try (this also applies to the Litwiller

Duncan SSA theorem). Thus :

SSA Theorem for Non-Euclidean Geome try. // the first and second sides and the

angle opposite the first side of one per missible triangle are congruent to the first and second sides and the angle opposite the first side of another permissible triangle, respectively, such that the length of the first side is not less than^ that of the

second, then the triangles are congruent.

It should be noted that the same coun

terexample mentioned earlier (fig. 2) readily leads to a denial not only of the original SSA theorem in elliptic geometry but of each of the propositions A, B, C, and D used in its proof.

REFERENCES

Ay res, Frank. "Theory and Problems of Plane and

Spherical Trigonometry." In Schaum's Outline

Series, pp. 93-94. New York: McGraw-Hill, 1954.

Coxford, Arthur F., and Zalman P. Usiskin. Geome

try: A Transformation Approach. New York: Laid law Brothers, 1975.

(Continued on page 347)

May 1983 367



5. How do the operations described in items 3 and 4 affect the magic difference for a magic triangle?

6. What is the maximum or the mini mum difference obtainable from subtrac tive magic triangles of a given order?

7. Can subtractive magic triangles be formed from arithmetic sequences other than the natural numbers?

Clearly, questions such as these can be

gainfully explored by mathematics class es. To encourage further inquiry, students could also be asked to formulate a defini tion for the adding of subtractive magic triangles.

APPENDIX 1. Yes. Here are two examples:

7 6 4 9

4 2 2 8 5 1 3 6 5 1 3

d = 7 d = 1

2. Yes, if 2 ^ < 6. For > 6, it's an open question. Magic triangles of order 3 and 4 have

already been presented. Some triangles of order 5, 6, and 7 follow:

13 2 10

3 16 17 14

4 15 6 5

7 8 12 1 18 11 9 d= 34

12 2 13 12

5 6 2 9 10 14 3 10

4 3 5 11 897 1 15 11 4671 8

d= 19 d= 16

3. Yes

4. The answer is yes if the triangle is of even order. A magic triangle of odd order retains its magic property only if the constant k is positive.

5. Assume that we are dealing with a magic trian

gle of order n. From our answers to previous ques tions, it follows that?

a) Adding a constant k only results in a new magic triangle whose magic difference is d + k\

b) If either is even or if is odd and k is positive, then multiplying by k will result in a new magic triangle with a magic difference of d\k\.

The result of multiplying a subtractive magic trian

gle of odd order by a negative number may not be a

magic triangle.

6. Let be the order of the triangle. Then,

?<</<-|(5n-6) if is even, and

4 4 if is odd. These are only upper and lower bounds for d; they may not be the least upper or the greatest lower bounds for d.

7. Yes, provided that the given sequence is an

increasing one.

AN IMPROVEMENT ON SSA CONGRUENCE FOR GEOMETRY AND TRIGONOMETRY

{Continued from page 367)

Heath, Sir Thomas L., ed. The Thirteen Books of Euclid's Elements, vol. 1. New York: Dover, 1956.

Kay, David C. College Geometry. New York: Holt, Rinehart & Winston, 1969.

Litwiller, Bonnie H., and David R. Duncan. "SSA: When Does It Yield Triangle Congruence?" Math ematics Teacher 74 (February 1981):106-8.

School Mathematics Study Group (SMSG). Geome

try, as cited in Anita Tuller, A Modern Introduction to Geometries. New York: D. Van Nostrand Co., 1966.

Yeshurun, Shraga. "The Angle: A Logical Gap in

Teaching Geometry and Trigonometry and Its

Remedy." International Journal of Mathematics Edu cation in Science and Technology 13 (1982): 133-38.

Yeshurun, Shraga, and Ely Merzbach. "Mathematical-Educational Implications of Some Statistical Distribution Problems." International Journal of Mathematics Education in Science and

Technology (in press).

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AN IMPROVEMENT ON SSA CONGRUENCE FOR GEOMETRY AND TRIGONOMETRY