In the examples above, you used a process called inductive reasoning to continue the pattern. Inductive reasoning is.


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1 Lesson 7 Inductive ing 1. I CAN understand what inductive reasoning is and its importance in geometry 3. I CAN show that a conditional statement is false by finding a counterexample Can you find the next item in each pattern? Monday, Wednesday, Friday, 3, 6, 9, 12, 15,,,, 0.4, 0.04, 0.004, , In the examples above, you used a process called inductive reasoning to continue the pattern. Inductive reasoning is A conjecture is Examples: complete each conjecture The quotient of two positive numbers is. The number of lines formed by four points, no three of which are collinear, is. The product of two odd numbers is. Average Whale Lengths in feet Length of female (ft) Length of male (ft)
2 To show that a conjecture is TRUE, you must prove it (more on that later). To show that a conjecture is FALSE, you have to find a counterexample, which is Inductive reasoning process: 1. Look for a. 2. Make a. 3. the conjecture or find a. Ex: Show that each conjecture is false by finding a counterexample For every integer n, the value of n 3 is positive. Two complementary angles are not congruent PRACTICE complete the following the best you can 1. Find the next item in each pattern: March, May, July, 1 2 3,,, Complete the conjecture: The product of two even numbers is. 3. Show that each conjecture is false by finding a counterexample: Three points on a plane always form a triangle. For any real number x, if x 2 1, then x 1.
3 Lesson 8 Conditional s 2. I CAN recognize and write conditional statements and determine their truth value 3. I CAN show that a conditional statement is false by finding a counterexample A conditional statement is a statement that can written in the form If, then_. Symbols: The hypothesis is. The conclusion is. Example 1: a. If it is raining, then the sidewalks are wet. Hypothesis: Conclusion: b. A number is a rational number, if it is an integer. Hypothesis: Conclusion: Even though some statements don t have the words If and then, they are still conditional statements, if one statement depends on the other. Write the statement in Ifthen form. A right triangle is a triangle with one right angle. If, then. We have earlyrelease days on Wednesdays. If, then. Every conditional statement that is made is either true or false. The only way a statement can be considered false is if the hypothesis is true, and the conclusion is false. p q p q *Remember, to show that something is false, you only need to provide one counterexample where the hypothesis is true, and conclusion is false.
4 Example 3: Determine whether true or false. 1. If the month is August, then the next month is September. 2. If two angles are acute, then they are congruent. 3. If 4 is prime, then The negation of statement p, written as. If the statement is It is raining, the negation of that statement would be. Negations are used to write related conditionals. Related Conditionals TERM STATEMENT SYMBOLS CONDITIONAL CONVERSE INVERSE CONTRAPOSITIVE Example 4: Write the converse, inverse, and contrapositive. Then find its truth value. TERM CONDITIONAL CONVERSE INVERSE CONTRAPOSITIVE STATEMENT If an animal is a cat, then it has four paws. Truth Value Notice that the some of the related conditionals have the same truth value. These are called statements. and and are always logically equivalent. are always logically equivalent.
5 Lesson 9 Deductive ing 4. I CAN understand what deductive reasoning is and its importance in geometry 5. I CAN determine the validity of conclusions using the Laws of Detachment and Syllogism Deductive reasoning is. In deductive reasoning, if given facts are true and you apply correct logic, the conclusion must be true. Two basic laws that are used to apply logic: Law of Detachment: Example 1: According to the Law of Detachment, is the conjecture valid? In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the World Series in Conjecture: The Red Sox won the World Series. If you are tardy 3 times, you must go to detention. George is in detention. Conjecture: George was tardy 3 times. If a student passes his classes, the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports. If it is Halloween, then Sheila wears a costume. Sheila is wearing a costume. Conjecture: Today is Halloween. If you want the best hamburger there is, then you go to Red Robin. Brandon went to Red Robin. Conjecture: Brandon had the best hamburger there is.
6 Law of Syllogism: Example 2: Use the Law of Syllogism to determine if the conjecture is valid Given If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Figure WXYZ is a kite. Conjecture: Figure WXYZ is a polygon. If a number is divisible by 2, then it is even. If a number is even, then it is an integer. A number is an integer. Conjecture: It is divisible by 2. Example 3: Use the Law of Syllogism to draw a conclusion If you attend GLHS, then you are a Blue Devil. If you are a Blue Devil, then you are awesome. Conclusion: If you attend GLHS, If it is autumn, the leaves change color. If it is October, then it is autumn. Conclusion:
7 Lesson 10 Deductive ing 6. I CAN write a definition as a biconditional 7. I CAN determine if a biconditional statement is true or false A biconditional statement is Write the biconditional as its conditional and converse. A. An angle is a right angle if and only if it measures 90. B. A solution is neutral iff it s ph is 7. Write the converse and biconditional from the conditional. A. If 5x 8 37, then x 9. B. If you live in Hell, then you live in Michigan. As you can see, sometimes a conditional and its converse are not both true. For a biconditional to be true, its conditional AND converse must be true. If either statement is false, then the biconditional is false. In other words, a biconditional statement must be true both ways that is read.
8 Write the conditional and converse from the biconditional. Then, determine if the biconditional is true or false. A. A rectangle has side lengths of 10 cm and 30 cm if and only if its area is 300 cm 2. B. y 2 5 y 25 C. An angle has a measure of 180 if and only if it is a straight angle. In geometry, biconditionals are used to write definitions. A definition is valid if it can written as a true biconditional. Write the definition of vertical angles as a biconditional. Vertical angles are two nonadjacent angles formed by intersecting lines. Write the definition of binomial as a biconditional. A binomial is an algebraic expression with exactly two terms.
9 Lesson 11 Introduction to Proofs 9. I CAN justify a statement using a property, definition, postulate, or theorem 10. I CAN write a deductive proof involving lines, segments, and angles In this lesson, we are going to demonstrate the process of doing a geometric proof. Each STATEMENT in a proof must follow logically from what has come before and must have a reason to support it. The REASON may be a piece of given information, a definition, a postulate, a property, or a previously proved theorem. The idea is straightforward start with one or more given facts, apply a logical chain of reasoning, and end with a conclusion EXAMPLE B is the midpoint of AC AB EF Prove: BC EF EXAMPLE Prove: and are complementary A C B and C are complementary A B
10 EXAMPLE m A 60 m B 2m A Prove: and are supplementary A B 1. m A 60 ; m B 2m A 2. m B 2(60 ) 3. m B m A m B m A m B A and B are supplementary. EXAMPLE X is the midpoint of AY Y is the midpoint of XB Prove: AX YB
11 Lesson 12 Proving Geometric Theorems 10. I CAN write a deductive proof involving lines, segments, and angles 11. I CAN prove the following theorems: Right Angle Congruence, Vertical Angles, and Linear Pair Right Angle Congruence Theorem are right angles Prove: 1 and Linear Pair Theorem are a linear pair Prove: 1 and 2 are supplementary 1 and Vertical Angles Theorem Prove: and 3 are vertical angles
12 Lesson 13 Two Lines Cut by a Transversal 12. I CAN recognize angle pairs when two lines are cut by a transversal A transversal is a line that intersects two coplanar lines at two different points. The transversal and the other two lines r and s always form eight angles. t Term Definition Example Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Sameside Interior Angles Ex 2: Classifying Pairs of Angles a. corresponding angles b. alt. exterior angles c. alt. interior angles d. sameside interior angles Ex. 3: Identifying Angle Pairs and Transversals a. 1 and 3 b. 2 and 6 c. 4 and 6
13 Lesson 14 Proving Parallel Lines Theorems 12. I CAN recognize angle pairs when two lines are cut by a transversal 13. I CAN prove and apply the following theorems: Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, and SameSide Interior Angles In the previous lesson, we have already made conjectures about what happens to the angle pairs when we have two parallel lines and a transversal. We are going to prove the theorems below. In order to prove these theorems, one of the relationships must be accepted without proof as a postulate. It can then be used to prove the other relationships. We are going to accept corresponding angles as a postulate. Proof of the ALTERNATE INTERIOR ANGLES THEOREM Prove: 1. 2 and 2 and and are alternate interior angles 3 are alternate interior angles 2 are vertical angles Proof of the SAMESIDE INTERIOR ANGLES THEOREM Prove: 8 and 9 are sameside interior angles 8 and 9 are supplementary and 9 are sameside interior angles 2. Corresponding Angles Postulate 3. m 7 m 9 4. Definition of a Linear Pair 5. 7 and 8 are supplementary 6. Definition of Supplementary Angles 7. m 8 m Definition of Supplementary Angles
14 Practice: Use the parallel line theorems to find the missing angle. Be sure to state which theorem/postulate you used. 1. Find m ABD. 2. Find m TUS. 3. Find m ABC. 4. Find the value of x.
15 Lesson 15 Proving Lines Parallel 12. I CAN recognize angle pairs when two lines are cut by a transversal 14. I CAN prove and apply the following theorems: Converse of Corresponding Angles, Converse of Alternate Interior Angles, Converse of Alternate Exterior Angles, and Converse of SameSide Interior Angles We have already established the angle relationships that occur when we have parallel lines. What about the converse of these theorems and postulate? Are they true? These have been proven you are going to write two of the proofs for homework. Proof of the Converse of the Alternate Interior Angles Theorem Prove: n m Whereas the previous lesson gave us parallel lines so that we could use the angle pair relationships to solve equations, this lessons gives us angle pair relationships so that we can determine if the lines are parallel or not. If one of the boxes on your Parallel Lines Theorems Converses sheet is satisfied, then the lines are PARALLEL. If one of the boxes on your Parallel Lines Theorems Converses sheet is contradicted, then the lines are NOT PARALLEL. If the given information doesn t fit at all into one of the boxes on your Parallel Lines Theorems Converses sheet, then there is NOT ENOUGH INFORMATION to tell if the lines are parallel or not. Use the three statements above to do the example problems that follow on the back
16 Determine if the following relationships prove that the lines are parallel. If so, you must have a theorem/postulate that supports it. If not, determine if there is not enough info or not parallel. 1. m 2 150, m m 1 m m 6 100, m m 3 45, m m 2 50, m 6 60
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