Using Properties to Simplify and Solve

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This unit lays the foundation for proving geometric relationships by developing the effective thinking needed to construct mathematically accurate logical arguments and proofs. Inductive reasoning is introduced as the type of thinking needed to analyze patterns and create conjectures. Counterexamples are used to prove some conjectures false. Different forms of logical statements are explored, including the converse, inverse, contrapositive, and biconditional statements. Counterexamples are used again to show that a logical statement is false. Logical statements are then used in defining basic geometric objects (i.e. points, lines, and planes), postulates (i.e. addition, etc), and properties (i.e. reflexive, symmetric, transitive). The understanding of segments is explored with constructions (i.e. copying of segments) and deductive reasoning is used to prove basic geometric statements, including those about segments. Two column proofs are introduced and a connection is made between this type of proof, and written justification of steps used in an algebra problem. Multiple types of proofs are used to prove statements about segment congruence, including two-column proofs and paragraph proofs. As an introduction to coordinate geometry, the distance formula and the midpoint are derived and organized as proofs. The definition of equidistant is given and illustrated in the coordinate plane using the distance formula.

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