Abstract
This chapter is devoted to the study of the usual sets of necessary and sufficient conditions for two triangles to be considered the same, in a sense it will soon be made precise. We also discuss here the important fifth axiom of Euclid (known as the axiom of parallels), as well as several interesting and important consequences of it, most notably the triangle inequality. Finally, in the last section of the chapter, several special types of quadrilaterals will make their first appearance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The reader with some previous knowledge of Euclidean Geometry will promptly notice that we do not list below the reflexive property of congruence of triangles. In this sense, whenever we refer, in a certain context, to two triangles, we will implicitly assume that they are distinct ones.
- 2.
A reflection along a line is the transformation of the plane that associates to each point its symmetric with respect to a fixed line. So, this problem opens the way to the study of reflections along lines as geometric transformations. We shall not pursue such a viewpoint here; instead, we refer the reader to the superb book [24].
- 3.
The case M = P will be dealt with in Problem 19, page 83.
- 4.
Namely: through two distinct given points there passes only one line; every line segment can be extended into a line; given a point and a line segment having this point as an end, there exists a circle centered at that point and having the given line segment as a radius; all right angles are equal.
- 5.
- 6.
After Leonhard Euler, Swiss mathematician of the eighteenth century. For more on Euler, see the footnote of page 124.
References
T.L. Heath, The Thirteen Books of Euclid’s Elements (Dover, Mineola, 1956)
E.E. Moise, Elementary Geometry from an Advanced Standpoint (Addison-Wesley, Boston, 1963)
I.M. Yaglom, Geometric Transformations I (MAA, Washington, 1962)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Caminha Muniz Neto, A. (2018). Congruence of Triangles. In: An Excursion through Elementary Mathematics, Volume II. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77974-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-77974-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77973-7
Online ISBN: 978-3-319-77974-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)