The Difference between Congruence and Equality

 Most often the words congruence and equality (or congruent and equal) create more confusion among beginning students in geometry. These two words have been used interchangeably as if they have the same meaning. The fact is that they are not the same. Congruence is a relationship of shapes and sizes, such as segments, triangles, and geometrical figures, while equality is a relationship of sizes, such as lengths, widths, and heights. Congruence deals with objects while equality deals with numbers. You don’t say that two shapes are equal or two numbers are congruent. Two objects are said to be congruent if one can be exactly superimposed on the other. Remember that all congruent objects have equal corresponding parts. In the figure above, the two triangles are congruent thus, △ABC≅△DEF. If they are congruent then the length of their corresponding sides or segments are equal thus, AB=DE, BC=EF, AC=DF; in addition, the measure of their corresponding angles (which is a real number) are equal, hence, m∠A=m∠D, m∠B=m∠E, m∠C=m∠F. In Fig.2 below, the two quadrilaterals are not congruent, even if their heights are equal; MO=NP=QS=RT. Take note that their widths are not equal; MN≠QR and OP≠ST. Congruence of two objects or shapes must be checked for the equality of their parts before concluding their congruence or the lack of it. In geometry, a shape such as a polygon can be translated (moved), rotated, and flipped over without losing its property (this is referred to as rigid motion)—the distances of its vertices and lengths of its sides remain unchanged.   Initial analysis of the two figures above may lead you to conclude that they are not congruent since if point G of the figure at the right is made to coincide with point B of the figure at the left, the other points will not coincide. The fact is, the two figures are symmetric or one is a mirror image of the other. To show that they are actually congruent, the figure at the right must be rotated and flipped over. Note: The figures above, except Fig. 2, animate every 20 seconds. If not, click the figure; a page opens in a new tab.