Congruency Theorems in the Geometry Curriculum of High Schools: An International Comparison
Keywords:math education, Geometry, postulates, theorems, congruent triangles, proofs of theorems, SAS and SSA conditions
This study investigates different approaches in contemporary math education. The author compares his experience in teaching Geometry in Russia and in the United States. The roles of Math and its purposes in the high school curriculum are discussed. The focus is made on theorems about congruent triangles. The set of congruency theorems can be arranged in different ways depending on the level of education and diversity of students.
The central theme of the study is an investigation of the theorems and postulates used to conclude triangles are congruent. Every triangle consists of three angles and three sides – a total of six elements. For triangles to be congruent all these elements must be congruent. Specific theorems require only three elements to be congruent to determine the congruency of triangles. These theorems are: SAS theorem (it compares two sides and one included angle), ASA and AAS theorems (they compare two angles and one side), SSS theorem (it compares three sides), HL theorem (a specific case for right triangles). The problem under consideration is the case of SAS theorem in which the congruent angles are not the included angles between congruent sides. The author describes different cases related to this issue and investigates conditions regarding the congruency of such triangles – SSA conditions. Finally, proposals are made about options to teach congruence theorems for regular high school mathematics classes as well as Pre-AP mathematics classes.