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Unit 2 
SIMILARITY, CONGRUENCE, AND PROOFS
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IN THIS UNIT, STUDENTS WILL BE EXPECTED TO:


Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. The first unit of Analytic Geometry involves similarity, congruence, and proofs. Students will understand similarity in terms of similarity transformations, prove theorems involving similarity, understand congruence in terms of rigid motions, prove geometric theorems, and make geometric constructions. During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs.
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.
Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of “same shape” and “scale factor” developed in the middle grades. These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent.
Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources.
CODES: The State Department of Education Defines FIVE BIG CONCEPTS for Unit 1 in Analytic Geometry. We have coded them, as listed below.
1.A  SIMILARITY, CONGRUENCE AND PROOFS
1.B  SIMILARITY PROOFS
1.C  CONGRUENCE > RIGID MOTION
1.D  GEOMETRIC THEOREM PROOFS
1.E  GEOMETRIC CONSTRUCTIONS (COMPASS AND STRAIGHTEDGE)
1.A  SIMILARITY, CONGRUENCE AND PROOFS
1.B  SIMILARITY PROOFS
1.C  CONGRUENCE > RIGID MOTION
1.D  GEOMETRIC THEOREM PROOFS
1.E  GEOMETRIC CONSTRUCTIONS (COMPASS AND STRAIGHTEDGE)
KEY IDEAS, STANDARDS, AND EXAMPLES

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