Lesson Plan : Proving Triangle Congruency

Teacher Name:
 Richard Martine
 Grade 9-10

 Proving Triangles congruent using Theorems and Postulates.
 Triangle congruency is proven through the use of one of five postulate and theorems available. Critical and logical thinking in solving congruency is based on using two-column proofs in order to display the thought process in a logical and orderly fashion.
 Students are to develop the skill of deductive logical reasoning in order to successfully solve the problems of proving triangles congruent. The student's logical reasoning must be expressed in the form of a two-column proof. Each decision must be justified by citing postulates, theorems, laws, or definitions.
 Using NC Objective 2.01, Students will use logic and deductive reasoning to draw conclusions and solve problems. Using NC Objective 2.03, Students will apply properties, definitions, and theorems of two-dimensional figures (triangles)to solve problems and write proofs.
 Geometry textbook, Geometry workbook, Overhead projector, View graphs.
 Students are presented with a representation of two triangles. Although they appear identical, students must come up with absolute reasons why the triangles are identical (congruent). Students must list the reasons with supporting postulates, theorems, and definitions.
 In order for two triangles to be congruent, all three sides and all three angles of each triangle must be congruent. Several shortcuts allow students to prove triangle congruency without proving all six components are identical. They are SSS, SAS, ASA, AAS, and HL. Any two triangle relationships can be proven by knowing only three of the six complonens, such as side-angle-side or with a right triangle, only Hypotenuse and one leg of the triangle.
 Students are provided with a worksheet containing as many as fifteen sample problems to work for the remainder of the period (approximately 45 minutes. The instuctor frequently moves around the classroom while the students work independently, monitoring their work and answering questions as necessary. Students are allowed to compare answers and assist other students on a limited basis. Several of the problems are discussed to ensure all students are finding the correct solutions.
 All students are expected to successfully achieve the basic level of knowledge which is to identify congruency by one of the five shortcuts. Above average achievers should be able to successfully complete a fill in the blank two-column proof with either mathematical statements or justification for given statements. The overachieving student should be able to completely write a two-column proof listing all mathematical statements and justification in a logical format.
Checking For Understanding:
 Students will be required to undergo a pop quizzes (both oral and written), quizzes and unit tests and successfully demonstrate their ability to write proofs while proving triangle congruency.
 Proving triangle congruency through the use of two-column proofs will allow students to develop logical reasoning to solve problems of any nature, not only in Geometry, but in other math and science classes as well. The ability to apply deductive reasoning will assist students in solving complex problems encountered in every day life.
 Once becoming indoctrinated to the benefits of two-column proofs and logical reasoning applied to triangle congruency, the success of the lesson is constantly proven when students begin using the technique to solve other geometric problems.
Teacher Reflections:
 It took several similar lessons before the average student caught on to the technique. Those who are quick learners were able to use the proofs almost immediately to solve other more complex problems. The average students do well in creating proofs, but often do not write the steps in perfect order. Average students do well if the steps are presented in order with clues and fill in the blanks accordingly.

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