Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures 2-1 Using Inductive Reasoning to Make Conjectures Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Geometry

2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 1A: Identifying a Pattern January, March, May,... The next month is July. Alternating months of the year make up the pattern.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 1B: Identifying a Pattern 7, 14, 21, 28, … The next multiple is 35. Multiples of 7 make up the pattern.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Complete the conjecture. Example 2A: Making a Conjecture The sum of two positive numbers is ?. The sum of two positive numbers is positive. List some examples and look for a pattern = = , ,000,017 = 1,003,917

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Check It Out! Example 2 The product of two odd numbers is ?. Complete the conjecture. The product of two odd numbers is odd. List some examples and look for a pattern. 1  1 = 1 3  3 = 9 5  7 = 35

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. To show that a conjecture is always true, you must prove it. A counterexample can be a drawing, a statement, or a number.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 4A: Finding a Counterexample For every integer n, n 3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n 3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n 3 = –27 and –27  0, the conjecture is false. n = –3 is a counterexample.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 4B: Finding a Counterexample Two complementary angles are not congruent. If the two congruent angles both measure 45°, the conjecture is false. 45° + 45° = 90°

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Check It Out! Example 4b Supplementary angles are adjacent. Show that the conjecture is false by finding a counterexample. The supplementary angles are not adjacent, so the conjecture is false. 23 ° 157 °

Holt McDougal Geometry 2-2 Conditional Statements 2-2 Conditional Statements Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Geometry

2-2 Conditional Statements By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.

Holt McDougal Geometry 2-2 Conditional Statements Identify the hypothesis and conclusion of each conditional. Example 1: Identifying the Parts of a Conditional Statement A.If today is Thanksgiving Day, then today is Thursday. B. A number is a rational number if it is an integer. Hypothesis: Today is Thanksgiving Day. Conclusion: Today is Thursday. Hypothesis: A number is an integer. Conclusion: The number is a rational number.

Holt McDougal Geometry 2-2 Conditional Statements Check It Out! Example 1 "A number is divisible by 3 if it is divisible by 6." Identify the hypothesis and conclusion of the statement. Hypothesis: A number is divisible by 6. Conclusion: A number is divisible by 3.

Holt McDougal Geometry 2-2 Conditional Statements Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other.

Holt McDougal Geometry 2-2 Conditional Statements Write a conditional statement from the following. Example 2A: Writing a Conditional Statement An obtuse triangle has exactly one obtuse angle. If a triangle is obtuse, then it has exactly one obtuse angle. Identify the hypothesis and the conclusion. An obtuse triangle has exactly one obtuse angle.

Holt McDougal Geometry 2-2 Conditional Statements Check It Out! Example 2 Write a conditional statement from the sentence “Two angles that are complementary are acute.” If two angles are complementary, then they are acute. Identify the hypothesis and the conclusion. Two angles that are complementary are acute.

Holt McDougal Geometry 2-2 Conditional Statements A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false.

Holt McDougal Geometry 2-2 Conditional Statements Determine if the conditional is true. If false, give a counterexample. Example 3A: Analyzing the Truth Value of a Conditional Statement If this month is August, then next month is September. When the hypothesis is true, the conclusion is also true because September follows August. So the conditional is true.

Holt McDougal Geometry 2-2 Conditional Statements Determine if the conditional is true. If false, give a counterexample. Example 3B: Analyzing the Truth Value of a Conditional Statement You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false. If two angles are acute, then they are congruent. Since you can find a counterexample, the conditional is false.

Holt McDougal Geometry 2-2 Conditional Statements DefinitionSymbols A conditional is a statement that can be written in the form "If p, then q." p  qp  q Related Conditionals

Holt McDougal Geometry 2-2 Conditional Statements DefinitionSymbols The converse is the statement formed by exchanging the hypothesis and conclusion. q  pq  p Related Conditionals

Holt McDougal Geometry 2-2 Conditional Statements DefinitionSymbols The inverse is the statement formed by negating the hypothesis and conclusion. ~p  ~q Related Conditionals

Holt McDougal Geometry 2-2 Conditional Statements DefinitionSymbols The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. ~q  ~p Related Conditionals

Holt McDougal Geometry 2-2 Conditional Statements Write the converse, inverse, and contrapostive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each. Check It Out! Example 4 If an animal is a cat, then it has four paws.

Holt McDougal Geometry 2-2 Conditional Statements Check It Out! Example 4 Inverse: If an animal is not a cat, then it does not have 4 paws. Converse: If an animal has 4 paws, then it is a cat. Contrapositive: If an animal does not have 4 paws, then it is not a cat; True. If an animal is a cat, then it has four paws. There are other animals that have 4 paws that are not cats, so the converse is false. There are animals that are not cats that have 4 paws, so the inverse is false. Cats have 4 paws, so the contrapositive is true.

Holt McDougal Geometry 2-2 Conditional Statements Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures 2-3 Using Deductive Reasoning to Verify Conjectures Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Geometry

2-3 Using Deductive Reasoning to Verify Conjectures Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Is the conclusion a result of inductive or deductive reasoning? Example 1A: Media Application 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. Since the conclusion is based on a pattern of observations, it is a result of inductive reasoning.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Is the conclusion a result of inductive or deductive reasoning? Example 1B: Media Application 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. The conclusion is based on logical reasoning from scientific research. It is a result of deductive reasoning.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The Law of Detachment is one valid form of deductive reasoning. Law of Detachment If p  q is a true statement and p is true, then q is true.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Determine if the conjecture is valid by the Law of Detachment. Example 2A: Verifying Conjectures by Using 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. The area of ∆PQR is 30 cm 2. Conjecture: The side lengths of ∆PQR are 5cm, 12 cm, and 13 cm.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures The given statement “The area of ∆PQR is 30 cm 2 ” matches the conclusion of a true conditional. But this does not mean the hypothesis is true. The dimensions of the triangle could be different. So the conjecture is not valid. Example 2A: Verifying Conjectures by Using the Law of Detachment Continued Identify the hypothesis and conclusion in the given conditional. 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.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Determine if the conjecture is valid by the Law of Detachment. Example 2B: Verifying Conjectures by Using the Law of Detachment Given: In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series. Conjecture: The Red Sox won the 2004 World Series.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Example 2B: Verifying Conjectures by Using the Law of Detachment Continued Identify the hypothesis and conclusion in the given conditional. In the World Series, if a team wins four games, then the team wins the series. The statement “The Red Sox won four games in the 2004 World Series” matches the hypothesis of a true conditional. By the Law of Detachment, the Red Sox won the 2004 World Series. The conjecture is valid.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Another valid form of deductive reasoning is the Law of Syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Law of Syllogism If p  q and q  r are true statements, then p  r is a true statement.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Determine if the conjecture is valid by the Law of Syllogism. Example 3A: Verifying Conjectures by Using 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.

Holt McDougal Geometry 2-3 Using Deductive Reasoning to Verify Conjectures Example 3A: Verifying Conjectures by Using the Law of Syllogism Continued Let p, q, and r represent the following. p: A figure is a kite. q: A figure is a quadrilateral. r: A figure is a polygon. You are given that p  q and q  r. Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that p  r. The conjecture is valid by Law of Syllogism.