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More About Dual Spaces
Published in James K. Peterson, Basic Analysis III, 2020
Comment 11.1.1 Thus J: X → X″ is what is called a congruence between X and J(X) ⊂ X″. A congruence is thus a linear map which is a norm preserving bijection. The symbol we use for congruence is ≅; so we would say X ≅ J(X).
Additional Topics in Digital Design
Published in Joseph Cavanagh, Digital Design and Verilog HDL Fundamentals, 2017
Congruence is indicated either by the symbol ≡ or by the symbol ⇨. The residue R of a number A modulo-m is specified by Equation 12.16. The integers A and B are of the form shown in Equation 12.17, where ai and bi are the digit values and r is the radix.
Angles and triangles
Published in John Bird, Basic Engineering Mathematics, 2017
Two triangles are said to be congruent if they are equal in all respects; i.e. three angles and three sides in one triangle are equal to three angles and three sides in the other triangle. Two triangles are congruent ifthe three sides of one are equal to the three sides of the other (SSS),two sides of one are equal to two sides of the other and the angles included by these sides are equal (SAS),two angles of the one are equal to two angles of the other and any side of the first is equal to the corresponding side of the other (ASA), ortheir hypotenuses are equal and one other side of one is equal to the corresponding side of the other (RHS).
Students' conceptions of the definitions of congruent and similar triangles
Published in International Journal of Mathematical Education in Science and Technology, 2022
The concept of congruent triangles is an important part of the basic knowledge needed to teach plane geometry (Patkin & Plaksin, 2011). The congruent triangle has a significant position because it links to similarity, because if two triangles are congruent, then they are also similar with a ration corresponding sides of 1:1. Furthermore, the three conditions for triangle congruency also serve as the basis for proving other propositions (Jones et al., 2013). Wu (2005) claimed that the cases of congruency and similarity highlight the need for definitions; without a mathematical definition of congruence and without a precise definition of similarity, learners cannot properly understand other topics in geometry, such as length and area.