Section 2.1-Inductive Reasoning And Conjecture - Mr. Naughton's Math Class

Section 2.1-Inductive Reasoning and Conjecture Definitions Inductive Reasoning- Conjecture- Counterexample- Examples 1-6: Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequences. 1. Costs: 4.50, 6.75, 9.00 . . . 2. Appointment times: 10:15 am, 11:00 am, 11:45 am . . . 3. . . . 4. 3, 3, 6, 9, 15 . . . 5. 2, 6, 14, 30, 62 . . . 6.

Examples 7-10: Make a conjecture about each value or geometric relationship. 7. The product of two even numbers. 8. The relationship between a and b if a b 0. 9. The relationship between the set of points in a plane equidistant from point A. 10. The relationship between AP and PB if M is the midpoint of AB and P is the midpoint of AM . Examples 11-12: Find a counterexample to show that each conjecture is false. 11. If A and B are complementary angles, then they share a common side. 12. If a ray intersects a segment at its midpoint, then the ray is perpendicular to the segment.

Section 2.2 – Conditional Statements Term conditional Statement hypothesis conclusion converse inverse Contrapositive Biconditional Definitions Symbols

Logically equivalent- o o Examples 1-2: Identify the hypothesis and conclusion of each conditional statement. 1. If today is Friday, then tomorrow is Saturday. 2. If two angles are supplementary, then the sum of the measures of the angles is 180. Examples 3-5: Write each statement in if-then form. 3. Sixteen-year olds are eligible to drive. 4. Cheese contains calcium. 5. The measure of an acute angle is between 0 and 90.

Examples 6-7: Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. 6. If x2 16 , then x 4 7. If an animal is spotted, then, it is a Dalmatian. Examples 8-10: Rewrite each statement as a biconditional statement. Then determine whether the biconditional is true or false. 8. Two angles whose measures add up to 90 are complementary. 9. There is no school on Saturday. 10. Perpendicular lines meet at right angles.

Example 11: Write the a) converse, b) inverse, and c) contrapositive of: “Get a free milkshake with any combo purchase” a) b) c) Section 2.3 – Deductive Reasoning Deductive reasoning- o Law of Detachment- o Law of Syllogism-

Example of Law of Detachment: Example of Law of Syllogism: Examples 1-2: Determine whether each conclusion is based on inductive or deductive reasoning. 1. Students at Olivia’s high school must have a B average in order to participate in sports. Olivia has a B average, so she concludes that she can participate in sports at school. 2. Holly notices that every Saturday, her neighbor mows his lawn. Today is Saturday. Holly concludes that her neighbor will mow his lawn. Examples 3-4: Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. 3. Given: If a number is divisible by 4, then the number is divisible by 2. 12 is divisible by 4. Conclusion: 12 is divisible by 2. 4. Given: If Ava stays up late, she will be tired the next day. Ava is tired. Conclusion: Ava stayed up late.

Examples 5-6: Using the Law of Syllogism, determine whether each statement is valid based on the information. If not, write invalid. Explain your reasoning. If a triangle is a right triangle, then it has an angle that measures 90 If a triangle has an angle that measures 90 , then its acute angles are complementary. 5. If the acute angles of a triangle are complementary, then it is a right triangle. 6. If a triangle is a right triangle, then its acute angles are complementary. Example 7: Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no valid conclusion and explain your reasoning. 7. Given: If Dalila finishes her chores, then she will receive her allowance. If Dalila receives her allowance, she will buy a new CD.

Section 2.4 Part I – Writing Proofs Definitions Postulate (axiom)- Postulate Postulate 2.1- Postulate 2.2- Postulate 2.3- Postulate 2.4- Postulate 2.5- Postulate 2.6- Postulate 2.7- Diagram

Examples 1-2: Explain how the figure illustrates that each statement is true. The state the postulate that can be used to show each statement is true. 1. Planes P and Q intersect in line r. 2. Plane P contains the points A, F, and D. Examples 3-4: Determine whether each statement is always, sometimes, or never true. Explain your reasoning. 3. The intersection of 3 planes is a line. 4. Through 2 points, there is exactly 1 line. Example 5: In the figure, , is in plane P and M is on be used to show each statement is true. 5. N and K are collinear. Section 2.4 Day II – Algebra Proofs Definitions Deductive Argument- Algebraic proof- . State the postulate that can

Properties of Real Numbers Property Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality Distributive Property Example

Property Segments Reflexive Symmetric Transitive Examples 1-5: State the property that justifies each statement. 1. If 4 5 1 , then x 4 5 x 1 2. If 5 y then y 5 3. If m 1 m 2 and m 2 m 3 then m 1 m 3 4. XY XY 5. If 2 x 5 11 , then 2 x 6 Two-column proof (formal proof)- Example 6: Given: 6 x 2 x -1 30 Prove: x 4 Angles

Example 7: Given: 4x 6 9 2 Prove: x 3 Section 2.4 Part III – Algebraic Proof Example 1: If 4 x 3 5 x 24 , then x 12 Example 2: If AB CD , then x 7

Example 3: If DF EG , then x 10 D E 11 2x-9 F G Example 4: A B Given: B 2 C m C 45 Prove: m A 90 Section 2.5 Day 1 – Proving Segment Relationships POSTULATE LIST Postulate 2.8- Postulate 2.9-

Example 1: Given: BC DE Prove: AB DE AC C B E A D Example 2: Given: Q is between P and R. R is between Q and S. PR QS Prove: PQ RS P Q Example 3: Given: CE FE ED EG Prove: CD FG R S

Example 4: Example 4: Given: JL KM Prove: JK LM Section 2.5 Day 2 – Proving Segment Relationships TheoremTHEOREM LIST Theorem 2.1- Theorem 2.2- Properties of Segment Congruence Property Example Reflexive of Congruence Symmetric of Congruence Transitive of Congruence

Example 1: Given: AB DE B is the midpoint of AC E is the midpoint of DF Prove: BC EF A B C D E F Example 2: Given: GA RP Prove: GR AP G A R P

Example 3: Given: AB AC PC QB Prove: AP AQ Section 2.6 Day 1– Proving Angle Relationships POSTULATE LIST Postulate 2.10- Postulate 2.11-

THEOREM LIST Theorem 2.3- Theorem 2.4- Theorem 2.5- Properties of Angle Congruence Property Example Reflexive of Congruence Symmetric of Congruence Transitive of Congruence Theorem 2.6- Theorem 2.7- Examples 1-2: Find the measure of each numbered angle at the right 1. m 2 x m 3 x -16 2. m 4 2 x m 5 x 9

Examples 3-5: Write a two-column proof for each of the following 3. Given : 1 3 Prove : WXZ TXY T Z W Y 1 2 3 X 4. Given : 1 and 2 form a linear pair 2 and 3 are supplementary Prove : 1 3 1 2 3

⃗⃗⃗⃗⃗ bisects 𝐷𝐵𝐸 5. Given: 𝐵𝐶 Prove: 𝐴𝐵𝐷 𝐴𝐵𝐸 Section 2.6 Day 2– Proving Angle Relationships Theorem 2.8- Theorem 2.9- Theorem 2.10- Theorem 2.11- Theorem 2.12- Theorem 2.13-

Examples 1-4: Write a two-column proof for each of the following: 1. Given : 1 3 Prove : 1 2 1 2 3 2. Given : NO OR POR is comp to PRO Prove : NOP PRO

3. Given : V YRX Y TRV Prove : V Y 4. Given : 1 2 1 3 Prove : FH bisects EFG

Section 2.1-Inductive Reasoning and Conjecture Definitions Inductive Reasoning- Conjecture- Counterexample- Examples 1-6: Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequences. 1. Costs: 4.50, 6.75, 9.00 . . . 2.