In the module, Congruence, congruence was used to prove that the base angles of an isosceles triangle are equal. The axis of symmetry of an isosceles triangle We begin by relating the reflection and rotation symmetries of isosceles triangles, parallelograms and rectangles to the results that we proved in the previous module, Paralleograms and Rectangles. Symmetries of triangles, parallelograms and rectangles The tests for the kite also allow several important standard constructions to be explained very simply as constructions of a kite.The reflection and rotation symmetries of triangles and special quadrilateralsĪre identified and related to congruence.In addition, two other matters are covered in these notes.
This is typical of more advanced mathematics. This means that the reader must understand a whole ‘sequence of theorems’ to achieve some results. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.Ĭongruence is again the basis of most arguments concerning rhombuses, squares, kites and trapezia, because the diagonals dissect each figure into triangles.Ī number of the theorems proved in this module rely on one or more of the previous theorems in the module.If a quadrilateral is a parallelogram, then its diagonals bisect each other.For example, a typical property−test pair from the previous module is the pair of converse statements: We have seen that a test for a special quadrilateral is usually the converse of a property. Many of the key methods of proof such as proof by contradiction and the difference between a theorem and its converse arise in elementary geometry.Īs in the module, Parallelograms and Rectangles, this module first stresses precise definitions of each special quadrilateral, then develops some of its properties, and then reverses the process, examining whether these properties can be used as tests for that particular special quadrilateral. These analytic skills can be transferred to many areas in commerce, engineering, science and medicine but most of us first meet them in high school mathematics.Īpart from some number theory results such as the existence of an infinite number of primes and the Fundamental Theorem of Arithmetic, most of the theorems students meet are in geometry starting with Pythagoras’ theorem. Logical argument, precise definitions and clear proofs are essential if one is to understand mathematics.
0 kommentar(er)
