section 2.1-inductive reasoning and conjecture

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 Definitions Symbols

conditional Statement

hypothesis

conclusion

converse

inverse

Contrapositive

Biconditional

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 2 16x , then 4x

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 Diagram

Postulate 2.1-

Postulate 2.2-

Postulate 2.3-

Postulate 2.4-

Postulate 2.5-

Postulate 2.6-

Postulate 2.7-

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 . State the postulate that can

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-

Properties of Real Numbers

Property Example

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

Property Segments Angles

Reflexive

Symmetric

Transitive

Examples 1-5: State the property that justifies each statement.

1. If 4 5 1 , then 4 5 1x x

2. If 5 y then 5y

3. If 1 2m m and 2 3m m then 1 3m m

4. XY XY

5. If 2 5 11x , then 2 6x

Two-column proof (formal proof)- Example 6:

Given: 6 2 -1 30x x

Prove: 4x

Example 7:

Given: 4 6

92

x

Prove: 3x

Section 2.4 Part III – Algebraic Proof Example 1:

If 4 3 5 24x x , then 12x

Example 2:

If AB CD , then 7x

Example 3:

If DF EG , then 10x

Example 4:

Given: 2

45

A B

B C

m C

Prove: 90m A

Section 2.5 Day 1 – Proving Segment Relationships POSTULATE LIST Postulate 2.8- Postulate 2.9-

GF

ED

2x-911

Example 1:

Given: BC DE

Prove: AB DE AC

Example 2: Given:

Prove: PQ RS

Example 3: Given:

Prove: CD FG

AE

D

C

B

P SQ R

Q is between P and R.

R is between Q and S.

= PR QS

CE FE

ED EG

Example 4: Example 4:

Given: JL KM

Prove: JK LM

Section 2.5 Day 2 – Proving Segment Relationships

Theorem- THEOREM 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:

Prove: BC EF

Example 2:

Given: GA RP

Prove: GR AP

A

ED

CB

F

G RA P

B is the midpoint of

E is the midpoint of

AB DE

AC

DF

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- Theorem 2.6- Theorem 2.7- Examples 1-2: Find the measure of each numbered angle at the right

1. 2

3 -16

m x

m x

2.

4 2

5 9

m x

m x

Properties of Angle Congruence

Property Example

Reflexive of Congruence

Symmetric of Congruence

Transitive of Congruence

Examples 3-5: Write a two-column proof for each of the following 3.

4.

3

21

Given : 1 and 2 form a linear pair

2 and 3 are supplementary

Prove : 1 3

Given : 1 3

Prove : WXZ TXY

Z

W

T

Y

1 2 3

X

5. Given: 𝐵𝐶⃗⃗⃗⃗ ⃗ bisects ∠𝐷𝐵𝐸

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. 2.

1 2

3

Given : 1 3

Prove : 1 2

Given :

is comp to

Prove :

NO OR

POR PRO

NOP PRO

3. 4.

Given : V YRX

Y TRV

Prove : V Y

Given : 1 2

1 3

Prove : FH bisects EFG