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Schema: Recursive Rule Explicit Rule 1. Generate a sequence using the description, then write a rule to predict the n-th term: “ The first term in the sequence is 4 and each term is three more than twice the previous term.” 1.3 The Axiomatic System of Geometry Solve Problems (Formally) Theorems Definitions & Properties Postulates & Undefined Terms Read & do #1 - #3 (then check answers in your group) Properties of Equality Distributive Property of Multiplication over addition (and subtraction?) Read & do #4 Check the answer in your group Two Column Proof Try These A a) 1)Read Example 2 & do Try These B 2) then do #5 3) Stop before #6 Example 2 & Try These B: Schema: Provide a 2 column proof for solving this equation Formative B only if = then implies = then Symbolic Logic pq Example: pq: is used to represent if p, then q or p implies q p: a number is prime q: a number has exactly two divisors If a number is prime, then it has exactly two divisors. Symbolic Logic - continued ~ is used to represent the word Example 1: p: the angle is obtuse ~p: The angle is not obtuse “not” Note: ~p means that the angle could be acute, right, or straight. Example 2: p: I am not happy ~p: I am happy ~p took the “not” out- it would have been a double negative (not not) Observations: • Conditional statements can be either true or false. • To show that a conditional statement is true, you must provide (write) a proof. • To show that a conditional statement is false, you must describe a single example that shows the statement is not always true. • The example that shows a statement is false is a COUNTEREXAMPLE. DO #6 Forms of Conditional Statements Converse: Switch the hypothesis and conclusion (q p) pq If two angles are vertical, then they are congruent. qp If two angles are congruent, then they are vertical. 27 Forms of Conditional Statements Inverse: State the opposite of both the hypothesis and conclusion. (~p~q) pq : If two angles are vertical, then they are congruent. ~p~q: If two angles are not vertical, then they are not congruent. Forms of Conditional Statements Contrapositive: Switch the hypothesis and conclusion and state their opposites. (~q~p) pq : If two angles are vertical, then they are congruent. ~q~p: If two angles are not congruent, then they are not vertical. Do #7, then check the answers in your group Forms of Conditional Statements • Contrapositives are logically equivalent to the original conditional statement. • If pq is true, then qp is true. • If pq is false, then qp is false. Biconditional • When a conditional statement and its converse are both true, the two statements may be combined. • Use the phrase if and only if (sometimes abbreviated: iff) Statement: If an angle is right then it has a measure of 90. Converse: If an angle measures 90, then it is a right angle. Biconditional: An angle is right if and only if it measures 90. Biconditional • All definitions can be written as biconditional statements. • 𝑝 𝑞 𝑎𝑛𝑑 𝑞 𝑡𝑟𝑢𝑒 𝑝 𝑝 𝑞 Do #8,9,10 &Try These C HW: • Try These C • P30, Check your understanding #1-13