Lesson 2-4Lesson 2-4Deductive Deductive ReasoningReasoning

Ohio Content Standards

Ohio Content StandardsEstablish the validity of conjectures Establish the validity of conjectures about geometric objects, their about geometric objects, their properties and relationships by properties and relationships by counter-example, inductive and counter-example, inductive and deductive reasoning, and critiquing deductive reasoning, and critiquing arguments made by others.arguments made by others.

Ohio Content StandardsMake and test conjectures about Make and test conjectures about characteristics and properties (e.g., characteristics and properties (e.g., sides, angles, symmetry) of two-sides, angles, symmetry) of two-dimensional figures and three-dimensional figures and three-dimensional objects.dimensional objects.

Ohio Content StandardsMake, test and establish the validity Make, test and establish the validity of conjectures about geometric of conjectures about geometric properties and relationships using properties and relationships using counterexample, inductive and counterexample, inductive and deductive reasoning, and paragraph deductive reasoning, and paragraph or two-column proof.or two-column proof.

Deductive Reasoning

Deductive Reasoning

Uses facts, rules, definitions, Uses facts, rules, definitions, or properties to reach logical or properties to reach logical conclusions.conclusions.

Law of Detachment

Law of Detachment

If If pp qq is true and is true and pp is true, is true, then then qq is also true. is also true.

The following is a true conditional. The following is a true conditional. Determine whether each conclusion is Determine whether each conclusion is valid based on the given information. valid based on the given information.

Explain your reasoning.Explain your reasoning.

The following is a true conditional. The following is a true conditional. Determine whether each conclusion is Determine whether each conclusion is valid based on the given information. valid based on the given information.

Explain your reasoning.Explain your reasoning.If two segments are If two segments are

congruent and the second congruent and the second segment is congruent to a segment is congruent to a

third segment, then the first third segment, then the first segment is also congruent to segment is also congruent to

the third segment.the third segment.

The following is a true conditional. The following is a true conditional. Determine whether each conclusion is Determine whether each conclusion is valid based on the given information. valid based on the given information.

Explain your reasoning.Explain your reasoning.If two segments are If two segments are

congruent and the second congruent and the second segment is congruent to a segment is congruent to a

third segment, then the first third segment, then the first segment is also congruent to segment is also congruent to

the third segment.the third segment.

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:Conclusion

; :Given

The following is a true conditional. The following is a true conditional. Determine whether each conclusion is Determine whether each conclusion is valid based on the given information. valid based on the given information.

Explain your reasoning.Explain your reasoning.If two segments are If two segments are

congruent and the second congruent and the second segment is congruent to a segment is congruent to a

third segment, then the first third segment, then the first segment is also congruent to segment is also congruent to

the third segment.the third segment.

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and :Conclusion

; :Given

Law of Syllogism

Law of Syllogism

If If pp qq is true and is true and qq rr is is also true then also true then pp rr is also is also true.true.

Use the Law of Syllogism to determine Use the Law of Syllogism to determine whether a valid conclusion can be whether a valid conclusion can be

reached from each set of statements.reached from each set of statements.

Use the Law of Syllogism to determine Use the Law of Syllogism to determine whether a valid conclusion can be whether a valid conclusion can be

reached from each set of statements.reached from each set of statements.

(1) If Sallie attends the (1) If Sallie attends the prom, she will go with prom, she will go with Mark.Mark.

(2) Mark is a 17-year-old (2) Mark is a 17-year-old student.student.

Use the Law of Syllogism to determine Use the Law of Syllogism to determine whether a valid conclusion can be whether a valid conclusion can be

reached from each set of statements.reached from each set of statements.

(1) If Mel and his date eat at (1) If Mel and his date eat at the Peddler Steakhouse the Peddler Steakhouse before going to the prom, before going to the prom, they will miss the senior they will miss the senior march.march.

(2) The Peddler Steakhouse (2) The Peddler Steakhouse stays open until 10 p.m.stays open until 10 p.m.

Determine whether statement (3) Determine whether statement (3) follows from statements (1) and (2) by follows from statements (1) and (2) by the Law of Detachment or the Law of the Law of Detachment or the Law of Syllogism. If it does, state which law Syllogism. If it does, state which law was used. If it does not, write was used. If it does not, write invalid.invalid.

Determine whether statement (3) Determine whether statement (3) follows from statements (1) and (2) by follows from statements (1) and (2) by the Law of Detachment or the Law of the Law of Detachment or the Law of Syllogism. If it does, state which law Syllogism. If it does, state which law was used. If it does not, write was used. If it does not, write invalid.invalid.

(1) If the sum of the squares (1) If the sum of the squares of two sides of a triangle is of two sides of a triangle is equal to the square of the equal to the square of the third side, then the triangle third side, then the triangle is a right triangle. is a right triangle.

(2) For (2) For XYZXYZ, , ( (XYXY))22 + ( + (YZYZ))22 = ( = (ZXZX))22..

(3) (3) XYZXYZ is a right triangle. is a right triangle.

Determine whether statement (3) Determine whether statement (3) follows from statements (1) and (2) by follows from statements (1) and (2) by the Law of Detachment or the Law of the Law of Detachment or the Law of Syllogism. If it does, state which law Syllogism. If it does, state which law was used. If it does not, write was used. If it does not, write invalid.invalid.(1) If Ling wants to participate in (1) If Ling wants to participate in the wrestling competition, he will the wrestling competition, he will have to meet an extra three times have to meet an extra three times a week to practice.a week to practice.

(2) If Ling adds anything extra to (2) If Ling adds anything extra to his weekly schedule, he cannot his weekly schedule, he cannot take karate lessons.take karate lessons.

(3) If Ling wants to participate in (3) If Ling wants to participate in the wrestling competition, he the wrestling competition, he cannot take karate lessons.cannot take karate lessons.

Assignment:Assignment:

Pgs. 85 - 87 Pgs. 85 - 87 12-29 all, 12-29 all,

44-54 evens44-54 evens