Congruence

Congruence is a quality between two objects or figures that have the same shape and size but may not be in the same positions. It is related to two other important concepts: equality and similarity. The simplest way to understand congruence is through visual transformations like rotation, reflection, and translation. Finally, proving similarity and congruence through strict criteria can be achieved through various methods known as angle-angle (AA), hypotenuse-leg (HL), angle-side-angle (ASA), and angle-angle-aide (AAS) criteria.

Identical shapes are often turned in different directions, or seem to be acting like mirror images of each other, and this is known as congruence. If all the angles and sides of one triangle are congruent to the corresponding angles and sides of another triangle, then the triangles are congruent. On the other hand, if it is already known that two triangles are congruent, their corresponding parts are also going to be found congruent. These facts make up what is knows as the Corresponding Parts of Congruent Figures are Congruent (CPCFC) theorem. This CPCFC theorem is true for any figure. If two figures are congruent, then any pair of corresponding parts is also congruent.

It is very important when writing about congruence that the order of things is made clear. If one writes XYZ is congruent to DEF then X is congruent to D, Y is congruent to E, and Z is congruent to F. It also means that XY = DE, YZ = EF, and XZ = DF. This organized statement can then lead to the Segment Congruence Theorem, which states that any congruent figures will produce any pair of corresponding segments that are congruent.

Congruence can also refer to circles, angles, line segments, or just two sets of points. For circles, they are congruent if they simply have the same diameter. For angles, they are congruent if they simply have the same measure in degrees. For lines, they are congruent if they simply have the same length.

History of Congruence

Congruence has always been an essential part of the historical foundation of Euclid’s geometry. Congruence tests, or axioms, have been around for thousands of years. These axioms are still used today to construct triangles according to specific lengths on a side or the specific degrees of an angle.

Thales was a Greek philosopher living in Egypt who studied geometry under many priests and who designed a method for finding the heights of the pyramids. Thales was likely to be the first mathematician to write proofs like the ones used for angles and triangles, and he became known as the father of Greek mathematics and the first of the Seven Wise Men of Greece.

Thales discovered that the base angles in isosceles triangles are equal and that there is such as thing as ASA congruence in triangles which was later confirmed by Pythagorus in the fifth century BC. Today we know that numbers can be figured into the axioms to determine whether a triangle is congruent with another and the idea of moving one triangle to fit perfectly on top of the other is not considered a perfect way to test for congruence.

In 1793, Eli Whitney, the inventor of the cotton gin, was also credited with the invention of interchangeable parts. Whitney used the concept of congruence in objects to create a large number of guns for the government and the guns could be mass produced because the parts could be made to be both similar and congruent through geometrical calculations. Before then, parts were always hand-made individually and, while similar, no two guns had parts that measured exactly the same. The idea of interchangeable parts depends on the concept of congruence and makes mass production possible.

Congruence vs. Equality

Congruence and equality utilize similar concepts but are used in different ways. Equality is used for numerical values such as the length of segments, the measures of angles, and the value of a slope. It is also used for sets because two sets often contain the same members and this produces equality. Congruence is different because it is used for figures. Some sets also create figures, as in the case of an angle where three points can be said to be equal to another set of points, but these points also create two figures and the figures are congruent. Line segments, triangles, and circles can all be considered as sets of points or figures with the same shape and size creating equal sets or congruent figures. The symbol for showing equality of sets is and is the symbol for congruence. The symbol is used where two figures are not congruent.

Rotation, Translation, and Reflection

Although they aren’t perfect, when testing for congruence, these three transformations are often the easiest way to test. Turning a shape is referred to as rotation. Flipping a shape is referred to as reflection. Sliding a shape is referred to as translation. After the shape has been transformed and the sides and angles each match, congruence can be proven. Of course, these transformations themselves don’t prove congruence. They are only an initial step in determining the best guess about a shape.

Congruence vs. Similarity

Similarity is also different from congruence. Determining similarity vs. a more perfect case of congruence is done according to the three simple transformations discussed. Simply turn, flip, or slide one of the shapes into a position of another shape. When the shapes appear to have the same lengths on each side, same degrees on each angle, same overall size and area, then they are congruent. When they only look the same, matching up in all ways except that one is larger or smaller than the other, the shapes are not congruent but only similar. The symbol is used to indicate similarity. In Figure 2, the pairs of figures A, B, and C are each similar but only the D figures (in red) are both similar and congruent.

An example of calculating the difference between similarity and congruence comes from the angle-angle (AA) similarity criterion. It is used when two angles are congruent with two other angles on a different triangle but there isn’t enough information to determine if these triangles are congruent. The congruence of angles results in the two triangles being called similar but not congruent.

Other criteria can be used with angles and triangles to determine congruence. The HL criterion states that two right triangles are congruent if the hypotenuse and one corresponding leg are similar in each. If both are 90° right triangles and the hypotenuse and leg measure equal, then the triangles are congruent. Knowing the difference between the AA criterion and the HL criterion helps in understanding the difference between similarity and congruence. Figure 3 shows an example of two right triangles that are congruent through the use of the HL criterion.

AAS and ASA Congruence

As the proofs for congruence get more advanced, the next two for congruent triangles are the Angle-Angle-Side (AAS) and the Angle-Side-Angle (ASA) criteria. In the first criteria AAS, two triangles are considered congruent when their two pairs of matching angles and one pair of matching sides each measure the same in both triangles. The matching side on each triangle is always the side opposite one of the same pair of angles. This can be seen in Figure 4 where the green side AC is opposite the red angle B. The two pairs of red and blue corresponding angles and the pair of corresponding green sides (AC and FD) are equal. As required, the side that is chosen is opposite the same two angles and this means the triangles are AAS congruent.

In the Angle-Side-Angle (ASA) criteria, two angles and only one side are also known to be the same. The difference in this criteria is that the sides that are equal are right between the two angles that are equal. This is the difference between AAS and ASA. The side that is the same on both triangles is set between the two angles, even though they are also the same in both triangles and the two criteria are very similar. An example of this can be seen in Figure 5.

References

Alonso, Orlando B., and Joseph Malkevitch. "Classifying Triangles and Quadrilaterals." The Mathematics Teacher 106.7 (2013): 541.

McKellar, Danica. Girls Get Curves. New York: Penguin, 2012. Print.

Posamentier, Alfred S, and Robert L. Bannister. Geometry, Its Elements and Structure. Mineola, NY: Dover, 2014.

Salinas, Tracie McLemore, et al. "Exploring Quadrilaterals to Reveal Teachers' Use of Definitions: Results and Implications." Australian Senior Mathematics Journal 28.2 (2014): 50-59.

Wolfe, Harold E. Introduction to Non-Euclidean Geometry. Mineola, NY: Dover, 2012.