Triangle Don t Talk to Me or My Son Ever Again Congruent Triangles

Historically (and by historically I mean "in Euclid's Elements") the word "equal", when applied to geometric figures, meant "equal in magnitude". So for example:

• Euclid refers to two segments equally equal if they accept the same length
• Two triangles are equal if they have the same area
• 2 solids are equal if they have the same book

For example, Elements Book I, Proposition 35 says:

Parallelograms which are on the same base of operations and between the aforementioned parallels are equal to one another.

(Meet figure below, from Fitzpatrick'south English translation of the Elements, based on the Greek text of Heiberg.)

Notice that "equal", in this usage, does not mean "coinciding"; nor does it mean "the same, identical". With this idea every bit background, the word "congruent" plays an of import role: it allows us to say that two figures are not but equal in area but are precisely the same shape, i.due east. can exist superimposed onto 1 some other.

Language, of course, is constantly evolving. At some indicate people stopped using the discussion "equal" to mean "equal in expanse", and began using it to mean "the aforementioned mathematical object". I am not sure exactly when this switch happened, but I suspect it was in the latter function of the 19th century, as gear up theoretical ideas began permeating all aspects of mathematics. The greatest body of water alter in Geometry during this time was Felix Klein's Erlangen Program, which sought to reframe Euclidean Geometry as the report of properties that are invariant under isometries of the plane. In this context it is of import to distinguish between the statements "$\triangle ABC = \triangle XYZ$" (which means that the ii triangles are the same mathematical object) and "$\triangle ABC \cong \triangle XYZ$" (which means that there exists an isometry mapping one triangle onto the other).

(This is as well when the partitional classification of quadrilaterals found in Euclid began to gave way to the hierarchical classification scheme most of us are more familiar with.)

The effect of this linguistic switch was that the relative valences of the words "equal" and "congruent" became reversed: whereas in Euclid'southward work the word "equal" is a weaker relation than the word "coinciding" (in that congruent figures are always equal, but the converse is non true), present the word "equal" describes a relationship stronger than congruence (in that "equal figures" are (trivially) congruent, only the antipodal is false.)

So in that location is i reason why it is important to accept unlike words: to distinguish between different notions of "same" ("aforementioned expanse" vs. "same effigy" vs. "same size and shape").

I would go even farther, and say that a lot of the vocabulary nosotros use in Geometry is introduced to disentangle notions that, to a naive pupil, seem equivalent.

• For case: to many students, a argument like "rectangle $ABCD$ is bigger than rectangle $PQRS$" seems perfectly sensible. That is considering they have an unexamined notion of "relative size". But information technology is possible for ane rectangle to have a larger diameter than some other, and simultaneously a smaller area, and equal perimeter! (Which is larger: a $half dozen \times 8$ rectangle, or a $vii \times 7$ rectangle?) So we introduce vocabulary (area, perimeter, diameter) in order to unpack these distinct notions of "size".
• A second, more sophisticated example: if you ask students to find a point in the interior of a triangle that is equidistant from all iii vertices, they will say "in the centre". If you ask students to find a point in the interior of a triangle that is equidistant from all 3 sides, they volition also say "in the center". It seems both obvious and unproblematic to them that "center of a triangle" means something -- and that it means one matter. But in fact there are multiple notions of "center", all of which are worth naming, and therefore we need dissimilar names (circumcenter, incenter, orthocenter, centroid) to distinguish them.

klinehably1937.blogspot.com

Source: https://matheducators.stackexchange.com/questions/20946/why-do-we-introduce-the-notion-that-triangles-are-congruent-instead-of-just-sa

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