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4. WARM-UPS Identify the hypothesis and conclusion of each statement. 1. If it rains on Monday, then I will stay home. 2.If x – 3 = 7, then x = 10. 3. If a polygon has six sides, then it is a hexagon. H: it rains on Monday; C: I will stay home H: x – 3 = 7; C: x = 10 H: a polygon has six sides; C: it is a hexagon

5. Contents Lesson 2-1Inductive Reasoning and Conjecture Lesson 2-2Logic Lesson 2-3Conditional Statements Lesson 2-4Deductive Reasoning Lesson 2-5Postulates and Paragraph Proofs Lesson 2-6Algebraic Proof Lesson 2-7Proving Segment Relationships Lesson 2-8Proving Angle Relationships

6. Lesson 3 Contents Example 4Related Conditionals OBJECTIVE: To write the converse, inverse, and contrapositive of if-then statements (2.8.11Q) (M8.D.1.1) KEY CONCEPTS: Conditional (p -> q); Converse (q->p); Inverse (~p->~q); Contra-positive (~q -> ~p)

7. Related Conditions pq Conditional p -> q Converse q -> p Inverse ~p -> ~q Contrapositive ~q -> ~p TTTTTT TFFTTF FTTFFT FFTTTT

8. Example 3-4a Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. First, write the conditional in if-then form. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. The converse is false. A rectangle with = 2 and w = 4 is not a square.

9. Example 3-4b Inverse: If a shape is not a square, then it is not a rectangle. The inverse is false. A 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square, but it is a rectangle. The contrapositive is the negation of the hypothesis and conclusion of the converse. Contrapositive: If a shape is not a rectangle, then it is not a square. The contrapositive is true.

10. Example 3-4b Write the converse, inverse, and contrapositive of the statement The sum of the measures of two complementary angles is 90. Determine whether each statement is true or false. If a statement is false, give a counterexample. Answer: Conditional: If two angles are complementary, then the sum of their measures is 90; true. Converse: If the sum of the measures of two angles is 90, then they are complementary; true. Inverse: If two angles are not complementary, then the sum of their measures is not 90; true. Contrapositive: If the sum of the measures of two angles is not 90, then they are not complementary; true.

11. Example 3-4b Write the conditional and converse of the statement The sum of the measures of two complementary angles is 90. Determine whether each statement is true or false. If a statement is false, give a counterexample. CONDITIONAL: If two angles are complementary, then the sum of their measures is 90; true. CONVERSE: If the sum of the measures of two angles is 90, then they are complementary; true.

12. Example 3-4b Write the inverse and contrapositive of the conditional statement: If two angles are complementary, then the sum of their measures is 90. Determine whether each statement is true or false. If a statement is false, give a counterexample. INVERSE: If two angles are not complementary, then the sum of their measures is not 90; true. CONTRAPOSITIVE: If the sum of the measures of two angles is not 90, then they are not complementary; true.

13. REVIEW Write the converse, inverse, and contrapositve of the following statement, then determine if each is true or false. If false, find a counterexample. If plants have water, then they will grow. CONVERSE: If plants grow, then they have water; true. INVERSE: If plants do not have water, then they will not grow; true. CONTRAPOSITIVE: If plants do not grow, then they do not have water. False; they may have been killed by overwatering.

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