Congruence (Triangle)
Congruence (Triangle)
Two triangles are congruent if they are alike in every geometric respect except, perhaps, one. That one possible exception is in the triangle’s “handedness.” There are only six parts of a triangle that can be seen and measured: the three angles and the three sides. The six features of a triangle are all involved with congruence. If triangle ABC is congruent to triangle DEF, then
Thus triangles ABC and DEF in Figure 1 are (or appear to be) congruent. (The symbol for “is congruent to” is ≌.)
The failure of congruence to include handedness usually does not matter. If triangle DEF were a
mirror, however, it would not fit into a frame in the shape of triangle ABC.
The term congruent comes from the Latin word congruere, meaning “to come together.” It therefore carries with it the idea of superposition, the idea that one of two congruent figures can be picked up and placed on top of the other with all parts coinciding. In the case of congruent triangles, the parts would be the three sides and the three angles.
Some authors prefer the word “equal” to “congruent.” Congruence is usually thought of as a relation between two geometric figures. In most practical applications, however, it is not the congruence of two triangles that matters, but the congruence of the triangle with itself at two different times.
There is a remarkably simple proof, for instance, that the base angles of an isosceles triangle are equal, but it depends on setting up a correspondence of a triangle ABC with the triangle ACB, which is, of course, the same triangle.
Two triangles are congruent if two sides and the included angle of one are congruent to two sides and the included angle of the other. This can be proven by superimposing one triangle on the other: they have to match. Therefore the third side and the two other angles have to match. Modern authors typically make no attempt to prove it, taking it as a postulate instead.
Whatever its status in the logical structure, side-angle-side congruence (abbreviated SAS) is a very useful geometric property. The compass with which one draws circles works because it has legs of a fixed length and a tight joint between them. In each of the positions the compass takes, the spacing between the legs—the third side of the triangles—is unchanging. Common shelf brackets support the shelf with two stiff legs and a reinforced corner joining them. A builder frames a door with measured lengths of wood and a carpenter’s square to make the included angle a right angle. In all these instances, the third side of the triangle is missing, but that does not matter. The information is sufficient to guarantee the length of the missing side and the proper shape of the entire triangle.
Two triangles are congruent if two angles and the included side of one are congruent respectively to two angles and the included side of the other. This is known as angle-side-angle (ASA) congruence, and is usually proved as a consequence of SAS congruence.
This is also a very useful property. Range finders, for example, used by photographers, golfers, and artillery observers, among others are based on ASA congruence. In Figure 2, the user sights the target C along line AC, and simultaneously adjusts angle B so that C comes into view along CB (small mirrors at B and at A direct the ray CB along BA and thence into the observer’s eye). The angle CAB is fixed; the distance AB is fixed; and the angle CAB, although adjustable, is measurable. By ASA congruence there is enough information to determine the shape of the triangle.
Although ASA congruence calls for an included side, the side can be an adjacent side as well. Since the sum of the angles of a triangle is always a straight angle, if any two of the angles are given, then the third angle is determined. The correspondence has to be kept straight, however. The equal sides cannot be an included side in one triangle and an adjacent side in the other or adjacent sides of angles which are not the equal angles.
Two triangles are congruent if three sides of one are equal, respectively, to the three sides of the other. This is known as side-side-side (SSS) congruence.
Probably the most widely exploited case of congruence is a folding ladder that is kept steady by a locking brace. Only the lengths of the three sides (see Figure 3) are fixed. The angles at A, B, and C are free to vary, and do so when the brace is unlocked and the ladder folded.
Draftsmen can replicate triangles easily and accurately with only a T-square, compass, and scale. Figure 4 shows the technique used. A contractor, with nothing more than a tape, some pegs, and some string, can lay out a rectangular foundation.
One set of criteria for congruence that can be used with caution is side-side-angle congruence. Figure 5 illustrates this. Here the lengths AB and AC are given. So is the size of angle C, which is not an included angle. With AB given, the vertex B can lie anywhere on a circle with center at A. If the second side of angle C misses the circle, no triangle meeting the specifications is possible. If it is tangent to the circle, one triangle is possible; if it cuts the circle, two are possible. This type of congruence comes into play when using the law of sines
to find an unknown angle, say angle B, by solving for sin B. If sin B is greater than 1, no such angle exists. If it equals 1, then B is a right angle. If it is less than 1, then sin B = sin (180– B) gives two solutions.
Resources
BOOKS
Euclid. Elements. Translated by Heath, Sir Thomas L. New York: Dover Publications, 1956.
Hahn, Liang-shin. Complex Numbers and Geometry. 2nd ed. The Mathematical Association of America, 1996.
Hilbert, D., and S. Cohn-Vossen. Geometry and the Imagination. New York: Chelsea Publishing Co. 1952.
Moise, Edwin E. Elementary Geometry from an Advanced Standpoint. Reading, Massachusetts: Addison-Wesley Publishing Co., 1963.
OTHER
Calkins, Keith G. “A Review of Basic Geometry—Lesson 7: Triangle Congruences. Not!” Andrews University. <http://www.andrews.edu/calkins/math/webtexts/geom07.htm> (accessed October 7, 2006.).
J. Paul Moulton