Volume and Surface Area of Solids -...

Inductive Reasoning “Courage is resistance to fear, mastery of

fear, not absence of fear.” –Mark Twain

Inductive Reasoning

O Inductive Reasoning is the process of observing a

pattern and making a conjecture about the pattern.

O A conjecture is an unproven statement based on

observations.

O So, inductive reasoning is the process of observing

a pattern and making an unproven statement

about the pattern.

O Ex. 1: Describe the pattern and either draw the next

shape or write the next three numbers.

1) 5, 15, 45, 135, …..

1) 405, 1215, 3645

2) -2, 1, 4, 7, ……

10, 13, 16

3)

4)

O Ex. 2: Make and test conjectures for each problem below.

1) The sum of two odd numbers.

3+5=8 7+9=16 -5+7=2

The sum of two odd numbers is even.

1) The product of two even numbers.

4x6=24 8x8=64 -4x4=-16

The product of two even numbers is even.

1) The sum of three consecutive odd numbers.

3+5+7=3(5) 1+3+5=3(3) 15+17+19=3(17)

The sum of three consecutive odd numbers is three times the middle number.

Counter-example

O A counter-example is the case that shows a

conjecture to be false.

O Conjecture: The product of any two numbers

is always positive.

O Counter-example: -5 x 4 = -20

O Conjecture: We have school each weekday.

O Counter-example: October 10th

O Ex. 3: Find the counter example to each conjecture.

1) All prime numbers are odd.

1) 2

2) If the sum of two numbers is even, then both of

those numbers are also even.

1) 3+5=8

1) The sum of three consecutive numbers is always

odd.

1) 3+4+5=12

Summary

O You should now be able to:

O Identify patterns and make conjectures about

those patterns.

O Provide counter-examples to false

conjectures.

Conditional Statements

“The great use of life is to spend it for

something that will outlast it.” –William

James

Conditional statements

O Each conditional statement has a condition and a

consequence.

O Hypothesis: is the condition of the statement (also

the if portion of the if-then format).

O Conclusion: is the consequence of the statement

(also the then portion of the if-then format)

O Example: All mammals have hair.

O If-then form: If an animal is a mammal, then it has

hair.

Ex. 1: For each conditional statement, write it in if-then form.

1) Every student at Westfield has to take Physical Science

their freshman year.

1) If a student is a freshman at Westfield, then they have to

take Physical Science.

2) Two angles are complementary if their measures add up

to 90°.

1) If two angle measures sum to be 90°, then they are

complementary angles

3) Vertical angles have two pairs of opposite rays.

1) If a pair of angles are vertical angles, then they have two

pairs of opposite rays.

4) 2x+5=2, because x=-6

1) If x=-6, then 2x+5=2

Negation

O A negation is the opposite of the original

statement.

O The apple is red.

O Negation: the apple is NOT red.

Converse, Inverse, and Contrapositive.

O Conditional statement: If I forget to put my name on a paper, then I get a zero for that paper.

O Converse: switch the hypothesis and conclusion of the original conditional statement.

O If I get a zero for a paper, then I forget to put my name on the paper.

O Inverse: Negate BOTH the hypothesis and the conclusion of the original conditional statement.

O If I remember to put my name on a paper, then I will not get a zero for that paper.

O Contrapositive: Negate BOTH the hypothesis and the conclusion of the converse.

O If I get greater than a zero on a paper, then I remember to put my name on the paper.

Ex. 2: For each conditional statement, write its

converse, inverse, and contrapositive and decide each

statements truth value.

1) If you watch this video, then you take notes for

geometry.

1) Converse: If you take notes for geometry, then you

watch this video.

2) Inverse: If you don’t watch this video, then you don’t

take notes for geometry.

3) Contrapositive: If you don’t take notes for geometry,

then you don’t watch this video.

Equivalent and Biconditional

O Equivalent statements are statements that are both true or are both false. The conditional statement and contrapositive are always equivalent. The inverse and converse are always equivalent.

O Biconditional statements are statements that the original conditional statement and its converse are both true.

O For example: If two angle measures sum to be 90°, then they are complementary.

O The definition could be written as: Two angles are complementary if and only if (iff) their measures sum to 90°.

Ex. 3: For each conditional statement, write its

converse, inverse, and contrapositive and decide each

statements truth value. If both the conditional and its

converse are true, write a biconditional statement.

1) If four points are coplanar, then they lie in the same

plane.

1) Converse: If four points lie in the same plane, then

they are coplanar.

2) Inverse: If four points aren’t coplanar, then they don’t

lie in the same plane, .

3) Contrapositive: If four points don’t lie in the same

plane, then they aren’t coplanar.

4) Biconditional: Four points are coplanar iff they lie in

the same plane.

Summary

O At this point, you should be able to:

O Write a converse, inverse, and contrapositive

to a conditional statement.

O Know the requirements for a statement to be

biconditional.

O Know how to negate a statement.

Deductive Reasoning “Only the person who has faith in himself

is able to be faithful to others.”–Erich

Fromm

Deductive Reasoning

O Deductive reasoning uses facts, definitions,

properties, and laws of logic to form a logical

argument.

Laws of Logic

O Law of detachment: If the hypothesis of a

true conditional statement is true, then the

conclusion is also true.

O If the flipped method proves to improve

learning, then Mr. H will continue it for the

whole year.

O If Mr. H gets a haircut, then pigs can fly.

Law of Syllogism (dominoes)

For want of a nail the shoe was lost.

For want of a shoe the horse was lost.

For want of a horse the rider was lost.

For want of a rider the message was lost.

For want of a message the battle was lost.

For want of a battle the kingdom was lost.

O These statements could be combined to be:

For want of a nail, the kingdom was lost.

Law of Syllogism (dominoes)

O If q, then r.

O If r, then s.

O If q, then s.

Ex. 1: Use the Law of Detachment to make a valid

conclusion.

1) If two angles have the same measure, then they

are congruent. The measure of angle A is 90° and

the measure of angle B is 90°.

1) Angle A is congruent to angle B.

2) Pythagoras takes a nap at 4pm. It is 4pm on

Saturday.

1) Pythagoras is taking a nap.

Ex. 2: Use the Law of Syllogism to make a valid

conclusion.

1) If two angles are both right angles, then they have

the same measure. If two angles have the same

measure, then they are congruent.

1) If two angles are both right angles, then they are

congruent.

2) If Jesse get a job, then he can afford a car. If Jesse

can afford a car, then he buys a car.

1) If Jesse gets a job, then he buys a car.

Ex. 3: Determine whether each statement is the result

of inductive or deductive reasoning. Explain why.

1) For the last two weeks Mr. H has gone around

helping students during the class period. You

conclude that Mr. H will help students during the

class period on Monday.

1) Inductive because you are making a conjecture

based on previous observations.

2) The rule at work is that you have to work the full

week to get paid on Friday. You were paid on Friday.

Therefore, you went to all of your classes.

1) Deductive because you use rules and facts to make

a conclusion.

Summary

O You should be able to use the laws of logic

to make valid conclusions.

O You should be able to determine the

difference between deductive and inductive

reasoning.

Using Postulates and Diagrams

“A hero is no braver than an ordinary man

(or woman), but he (/she) is brave five

minutes longer.” –Ralph Waldo Emerson

Postulates 5-11

O 5: Through any two points there exists

exactly one line.

O 6: A line contains at least two points.

O 7: If two lines intersect, then their

intersection is exactly one point.

O 8: Through any three noncollinear points

there exists exactly one plane.

Postulates 5-11

O 9: A plane contains at least three

noncollinear points.

O 10: If two points lie in a plane, then the line

containing them lies in the plane.

O 11: If two planes intersect, then their

intersection is a line.

O Ex. 1: State the postulate illustrated by the

diagram.

1)

2) A A

C

A

B B

B

C

O Ex. 2: Use the diagram to write examples of

postulate 5 and 7.

Through points C and B there

is one line called line l.

Line DE and Line BF intersect

at point D.

O Ex. 3: Use the diagram to determine if each

statement is true or false.

1) Line AB lies in plane

R

2) Line FH lies in plane

R

3) Line AC and Line FG

will intersect.

4) Line GH is

perpendicular to

plane R.

5) Angle LGH is a right

angle.

6) Angle LGH and

angle LGF are

supplementary

angles.

L

Summary

O You should be able to identify the postulate

used in drawing a diagram.

Reasoning using Algebra.

“Certain signs precede certain events.”

–Cicero

Algebraic Properties of Equality.

Let a, b, and c be real

numbers.

1) Addition Property

2) Subtraction Property

3) Multiplication Property

4) Division Property

5) Substitution Property

6) Distributive Property

1) If 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 = 𝑏 + 𝑐.

2) 𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 = 𝑏 − 𝑐.

3) 𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐 = 𝑏𝑐.

4) 𝐼𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑐 ≠ 0, 𝑡ℎ𝑒𝑛 𝑎

𝑐=

𝑏

𝑐.

5) 𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎 𝑐𝑎𝑛 𝑏𝑒 𝑠𝑢𝑠𝑡𝑖𝑡𝑢𝑑𝑒𝑑 𝑖𝑛 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑟

𝑒𝑥𝑝𝑟e𝑠𝑠𝑖𝑜𝑛.

6) 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐

Example 1: Solve 2𝑥 + 30 = 75 − 3𝑥. Write a

reason for each step.

Equation

2𝑥 + 30 = 75 − 3𝑥

5𝑥 + 30 = 75

5𝑥 = 45

𝑥 = 9

Reason

Given

Addition Property

Subtraction Property

Division Property

Example 2: Solve −2(𝑥 + 30) = 2(70 − 3𝑥). Write a

reason for each step.

Equation

−2(𝑥 + 30) = 2(70 − 3𝑥)

−2𝑥 − 60 = 140 − 6𝑥

4𝑥 − 60 = 140

4𝑥 = 200

𝑥 = 50

Reason

Given

Distributive Property

Addition Property

Addition Property

Division Property

Reflexive Properties of Equality.

1) Real Numbers

2) Segment Length

3) Angle Measure

1) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎, 𝑎 = 𝑎.

2) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵, 𝐴𝐵 = 𝐴𝐵

3) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑎𝑛𝑔𝑙𝑒 𝐴, 𝑚∠𝐴 = 𝑚∠𝐴.

Symmetric Properties of Equality.

1) Real Numbers

2) Segment Length

3) Angle Measure

1) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎 𝑎𝑛𝑑 𝑏, 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎.

2) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵 𝑎𝑛𝑑 𝐶𝐷, 𝑖𝑓 𝐴𝐵 = 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐶𝐷 = 𝐴𝐵.

3) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑎𝑛𝑔𝑙𝑒 𝐴 𝑎𝑛𝑑 𝐵, 𝑖𝑓 𝑚∠𝐴 = 𝑚∠𝐵, 𝑡ℎ𝑒𝑛 𝑚∠𝐵 = 𝑚∠𝐴.

Transitive Properties of Equality.

1) Real Numbers

2) Segment Length

3) Angle Measure

1) 𝐼𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐.

2) 𝐼𝑓𝐴𝐵 = 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 = 𝐸𝐹, 𝑡ℎ𝑒𝑛 𝐴𝐵 = 𝐸𝐹.

3) 𝐼𝑓 𝑚∠𝐴 = 𝑚∠𝐵 𝑎𝑛𝑑 𝑚∠𝐵 = 𝑚∠𝐶, 𝑡ℎ𝑒𝑛 𝑚∠𝐴 = 𝑚∠𝐶.

Example 3: Determine if 𝑚∠ABC= 𝑚∠𝐹𝐵𝐷. Show your reasoning.

Equation

𝑚∠1= 𝑚∠3

𝑚∠ABC= 𝑚∠1+ 𝑚∠2

𝑚∠FBD= 𝑚∠2 +𝑚∠3

𝑚∠FBD= 𝑚∠2+ 𝑚∠1

𝑚∠ABC= 𝑚∠FBD

Reason

Given

Angle Addition Postulate

Angle Addition Postulate

Substitution Property

Transitive Property

A

B

C

D

F

1 2

3

Summary

O You should be able to use properties to

justify your reasoning.

O You should be able to solve a problem and

provide reasons for each step.

Prove Statements about segments and

angles “Anxiety is fear of one’s self.” –Wilhelm

Stekel

Congruence of Segments and Angles. (Theorem 2.1 and 2.2)

1) Reflexive Property of Congruence

2) Symmetric Property of Congruence

3) Transitive Property of Congruence

For any segment AB and any angle A.

1) 𝐴𝐵 ≅ 𝐴𝐵 or ∠𝐴 ≅ ∠𝐴

2) If 𝐴𝐵 ≅ 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐶𝐷 ≅ 𝐴𝐵 or If ∠𝐴 ≅ ∠𝐵, 𝑡ℎ𝑒𝑛 ∠𝐵 ≅ ∠𝐴

3) 𝐼𝑓 𝐴𝐵 ≅ 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 ≅ 𝐸𝐹, 𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐸𝐹 or

𝐼𝑓 ∠𝐴 ≅ ∠𝐵 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝐶, 𝑡ℎ𝑒𝑛 ∠𝐴 ≅ ∠𝐶.

Example 1: Use a two column proof to show that AC ≅ 𝐵𝐷.

A B C D

Statements

1) 𝐴𝐵 = 𝐶𝐷

2) 𝐴𝐶 = 𝐴𝐵 + 𝐵𝐶

3) 𝐵𝐷 = 𝐶𝐷 + 𝐵𝐶

4) 𝐵𝐷 = 𝐴𝐵 + 𝐵𝐶

5) 𝐴𝐶 = 𝐵𝐷

6) AC ≅ 𝐵𝐷

Reasons

1) Given

2) Segment Addition Postulate

3) Segment Addition Postulate

4) Substitution Property

5) Transitive Property

6) Definition of Congruence

Ex. 2: Name the property illustrated by the statement.

1) 𝐼𝑓 ∠𝐹 ≅ ∠𝐺 𝑎𝑛𝑑 ∠𝐺 ≅ ∠𝐻, 𝑡ℎ𝑒𝑛 ∠𝐹 ≅ ∠𝐻.

2) If 𝐸𝐹 ≅ 𝐺𝐻, 𝑡ℎ𝑒𝑛 𝐺𝐻 ≅ 𝐸𝐹.

3) 𝐴𝐵 ≅ 𝐴𝐵

Example 1:Prove that AB=2𝐴𝑀. 𝑌𝑜𝑢 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑀 𝑖𝑠 𝑡ℎ𝑒

𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵.

A M B

Statements

1) 𝑀 𝑖𝑠 𝑎 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵. 2) 𝐴𝑀 ≅ 𝑀𝐵

3) 𝐴𝑀 = 𝑀𝐵 4) 𝐴𝐵 = 𝐴𝑀 + 𝑀𝐵

5) 𝐴𝐵 = 𝐴𝑀 + 𝐴𝑀

6) 𝐴𝐵 = 2𝐴𝑀

Reasons

1) Given

2) Definition of Midpoint

3) Definition of congruence

4) Segment Addition Postulate

5) Substitution Property

6) Simplify

Summary

O You should be able to prove statements

about segments and angles.

O You should be able to write a two column

proof.

Prove Angle Pair Relationships.

“Remember that happiness is a way of

travel—not a destination.” –Roy M.

Goodman

Theorem 2.3-5

O Theorem 2.3: Right Angles Congruence Theorem.

O All right angles are congruent.

O Theorem 2.4: Congruent Supplements Theorem.

O If two angles are supplementary to the same angle

(or to congruent angles), then they are congruent.

O Theorem 2.5: Congruent Complements Theorem.

O If two angles are complementary to the same angle

(or to congruent angles), then they are congruent.

1) Ex. 1: Prove that ∠1 ≅ ∠3, given that ∠1 𝑎𝑛𝑑 ∠2 are

supplementary and ∠3 𝑎𝑛𝑑 ∠2 are supplementary.

2

3 1

Statements

1) ∠1 𝑎𝑛𝑑 ∠2 are supp.

∠3 𝑎𝑛𝑑 ∠2 are supp.

2) 𝑚∠1 + 𝑚∠2 = 180

3) 𝑚∠3 + 𝑚∠2 = 180 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2

5) 𝑚∠1 = 𝑚∠3

6) ∠1 ≅ ∠3

Reasons

1) Given

2) Definition of Supplementary

3) Definition of Supplementary

4) Transitive Property

5) Subtraction Property

6) Definition of Congruence

Postulate 12 and Theorem 2.6

O Postulate 12: Linear Pair Postulate.

O If two angles form a linear pair, then they are

supplementary.

O Theorem 2.6: Vertical Angles Congruence Theorem.

O Vertical Angles are Congruent.

O Ex. 2: Given that angle 3 and angle 4 are a linear

pair and measure of angle 4 is 112°, find the

measure of angle 3.

Statements

1) ∠3 𝑎𝑛𝑑 ∠4 form a linear

pair and 𝑚∠4 = 112°

2) ∠3 𝑎𝑛𝑑 ∠4 are supp.

3) 𝑚∠3 + 𝑚∠4 = 180

4) 𝑚∠3 + 112 = 180 5) 𝑚∠3 = 68°

Reasons

1) Given

2) Linear Pair Postulate

3) Definition of Supplementary

4) Substitution Property

5) Subtraction Property

Ex. 3: Find the value of x if 𝑚∠1 = (3𝑥 − 4)° and 𝑚∠4 = (6𝑥 − 184)°.

1

2 3

4 5

Statements

1) 𝑚∠1 = 3𝑥 − 4 ° 2) 𝑚∠4 = (6𝑥 − 184)°

3) ∠1≅∠4

4) 𝑚∠1 = 𝑚∠4

5) 3𝑥 − 4 ° = (6𝑥 − 184)°

6) −4 = 3𝑥 − 184 7) 180=3x 8) 60=x

Reasons

1) Given

2) Given

3) Vertical Angle Congruence

Theorem

4) Definition of Congruence

5) Transitive Property

6) Subtraction Property

7) Addition Property

8) Division Property

Summary

O You should be able to identify

complementary and supplementary angles.

O You should be able to identify linear pairs

and vertical angles.

O You should be able to use the above

definitions, postulates, and theorems to

write a proof.