Unit 2: Geometric Reasoning Section 1: Inductive Reasoning Ex #1: Find the next item in the pattern. January, March, May,... Ex #2: Find the next item in the pattern. 7, 14, 21, 28, Ex #3: Find the next item in the pattern. Ex #4: Find the next item in the pattern 0.4, 0.04, 0.004, Inductive Reasoning: Ex #5: Complete the conjecture. The sum of two positive numbers is?. Ex #6: Complete the conjecture. The number of lines formed by 4 points, no three of which are collinear, is?. Ex #7: Complete the conjecture. The product of two odd numbers is?. 1
Inductive Reasoning Ex #8: Show that the conjecture is false by finding a counterexample. For every integer n, 3 n is positive. Ex #9: Show that the conjecture is false by finding a counterexample. Two complementary angles are not congruent. Ex #10: Show that the conjecture is false by finding a counterexample. 2 For any real number x, x x. Ex #11: Show that the conjecture is false by finding a counterexample. Supplementary angles are adjacent. 2
Unit 2: Geometric Reasoning Section 2: Conditional Statements Identify the hypothesis and conclusion of each conditional. Ex #1: A. If today is Thanksgiving Day, then today is Thursday. Ex #2: "A number is divisible by 3 if it is divisible by 6." Write a conditional statement from the following. Ex #3: An obtuse triangle has exactly one obtuse angle. Ex #4: Ex #5: Two angles that are complementary are acute. 3
Determine if the conditional is true. If false, give a counterexample. Ex #6: If this month is August, then next month is September. Ex #7: If two angles are acute, then they are congruent. Ex #8: If a number is odd, then it is divisible by 3 Write the converse, inverse, and contrapositive of the conditional statement. Use the Science Fact to find the truth value of each. Ex #9 If an animal is an adult insect, then it has six legs. Converse: Science Fact: Adult insects have six legs. No other animals have six legs. Inverse: Contrapositive: 4
Unit 2: Geometric Reasoning Section 3: Deductive Reasoning Deductive reasoning Ex #1: Is the conclusion a result of inductive or deductive reasoning? There is a myth that you can balance an egg on its end only on the spring equinox. A person was able to balance an egg on July 8, September 21, and December 19. Therefore this myth is false. Ex #2: Is the conclusion a result of inductive or deductive reasoning? There is a myth that the Great Wall of China is the only man-made object visible from the Moon. The Great Wall is barely visible in photographs taken from 180 miles above Earth. The Moon is about 237,000 miles from Earth. Therefore, the myth cannot be true. Ex #3: Is the conclusion a result of inductive or deductive reasoning? There is a myth that an eelskin wallet will demagnetize credit cards because the skin of the electric eels used to make the wallet holds an electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? 5
Ex #4: Determine if the conjecture is valid by the Law of Detachment. Given: In the World Series, if a team wins four games, then the team wins the series. Situation: The Red Sox won four games in the 2004 World Series. Conjecture: The Red Sox won the 2004 World Series. Ex #5: Determine if the conjecture is valid by the Law of Detachment. Given: If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm 2 Situation: The area of PQR is 30 cm. 2. Conjecture: The side lengths of PQR are 5cm, 12 cm, and 13 cm. Ex #6: Determine if the conjecture is valid by the Law of Detachment. Given: If a student passes his classes, the student is eligible to play sports. Situation: Ramon passed his classes. Conjecture: Ramon is eligible to play sports. 6
Ex #7: Determine if the conjecture is valid by the Law of Syllogism. Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Conjecture: If a figure is a kite, then it is a polygon. Ex #8: Determine if the conjecture is valid by the Law of Syllogism. Given: If a number is divisible by 2, then it is even. If a number is even, then it is an integer. Conjecture: If a number is an integer, then it is divisible by 2. Ex #9: Determine if the conjecture is valid by the Law of Syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair. Ex #10: Draw a conclusion from the given information. A. Given: If 2y = 4, then z = 1. If x + 3 = 12, then 2y = 4. x + 3 = 12 B. If the sum of the measures of two angles is 180, then the angles are supplementary. If two angles are supplementary, they are not angles of a triangle. m A= 135, and m B= 45. 7
Lesson Quiz: Is the conclusion a result of inductive or deductive reasoning? 1. At Reagan High School, students must pass Geometry before they take Algebra 2. Emily is in Algebra 2, so she must have passed Geometry. Determine if each conjecture is valid? 2. Given: If n is a natural number, then n is an integer. If n is an integer, then n is a rational number. 0.875 is a rational number. Conjecture: 0.875 is a natural number. 3. Given: If an American citizen is at least 18 years old, then he or she is eligible to vote. Anna is a 20-year-old American citizen. Conjecture: Anna is eligible to vote 8
Unit 2: Geometric Reasoning Section 4: Algebraic Proof Ex #1: Solve the equation 4m 8 = 12. Write a justification for each step. Ex #2: Solve the equation. Write a justification for each step. Ex #3: Write a justification for each step. Ex #4: Write a justification for each step. 9
REMEMBER: are equal (=) are congruent ( ). Ex #5: Identify the property that justifies each statement. A. QRS QRS B. m 1 = m 2 so m 2 = m 1 C. AB CD and CD EF, so AB EF. D. 32 = 32 Ex #6: Identify the property that justifies each statement. A. DE = GH, so GH = DE. B. 94 = 94 C. 0 = a, and a = x. So 0 = x. D. A Y, so Y A 10
Unit 2: Geometric Reasoning Section 5: Geometric Proof When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. Ex #1 Write a justification for each step, given that A and B are supplementary and m A = 45. 1. A and B are supplementary. m A = 45 2. m A + m B = 180 3. 45 + m B = 180 4. m B = 135 You Try #1 Write a justification for each step, given that B is the midpoint of AC and EF AB. 1. B is the midpoint of AC 2. EF AB 3. AB BC 4. EF BC 11
A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column. Ex #2 1. 1 and 2 form a linear pair. 1. Given 2. m ABC = 180 2. Def. of straight line 3. 3. Angle Addition Postulate 4. 4. Substitution Prop of = 5. 1 and 2 are supplementary 5. 12
You Try #2 1. 1. Given 2. m 1 + m 2 = 180 m 2 + m 3 = 180 2. Def. of supplementary s 3. 3. Substitution Prop of = 4. m 1 = m 3 4. 5. 5. Def. of s Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you. 13
Given: 1 and 2 are complementary, and 2 and 3 are complementary. Prove: 1 3 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 14