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Geometry October 20, 2016 CHAPTER 2 REVIEW DO NOW Solve for x. Write all of your steps clearly. 1. 2. Agenda • Announcements (2nd hour only) • Do Now • Review for Quiz • Exercises REMINDER •Chapter 2 Quiz TOMORROW! Objective/Big Question What does it mean to “prove” something in mathematics? Exercises from yesterday Conjecture • A prediction we make based on a pattern of numbers • PROVE: Use a formal PROOF to make a conjecture a THEOREM • DISPROVE: Find a single COUNTEREXAMPLE Conditional • An “if-then” statement: Something which states that, if A is true, then B is true. If today is Monday, we have to go to school. If it’s cold outside, you should wear a coat. If 3x = 12, then x = 4. Hypothesis • The “if” statement: What we are saying might be true. If today is Monday, we have to go to school. If it’s cold outside, you should wear a coat. If 3x = 12, then x = 4. Conclusion • The “then” statement: What logically follows from the hypothesis. If today is Monday, we have to go to school. If it’s cold outside, you should wear a coat. If 3x = 12, then x = 4. Euler Diagram • All whole numbers are integers • Conditional: If a number is a whole number, it is an integer. Biconditional • If a CONDITIONAL and its CONVERSE are both true, we can write it as a “biconditional”: HYPOTHESIS if and only if CONCLUSION • A number is prime if and only if it is greater than one and has no factors other than itself and one. • 5x = 20 if and only if x = 4 Definition • A DEFINITION is a BICONDITIONAL with a vocabulary word as its HYPOTHESIS: • A number is prime if and only if it is greater than one and has no factors other than itself and one. • An angle is right if and only if its measure is 90°. Proofs • Given: What you’re told is true • Prove: What you intend to show, using the given information and known properties, postulates, and theorems • Two-column proof: Each step of your proof is provided, with a stated reason for each step, in a table • Paragraph proof: Same steps, written as a paragraph Key Properties • REFLEXIVE PROPERTY: Everything is congruent to itself • SYMMETRIC PROPERTY: You can “flip” equalities and congruencies • TRANSITIVE PROPERTY: If two things are both congruent to a third thing, they’re congruent to each other • SUBSTITUTION: If two values are equal, you can swap them out Vertical Angles Theorem • VERTICAL ANGLES are always congruent • ∠1 ≅ ∠3 𝑚∠1 + 𝑚∠2 = 180° Quiz Review