2.5: conjectures that lead to theorems

04/19/2304/19/23 2.5: Conjectures that Lead to Theorems2.5: Conjectures that Lead to Theorems

2.5: Conjectures That Lead 2.5: Conjectures That Lead to Theorems to Theorems

Expectations:Expectations:G1.1.1: Solve multistep problems and construct proofs involving G1.1.1: Solve multistep problems and construct proofs involving

vertical angles, linear pairs of angles, supplementary vertical angles, linear pairs of angles, supplementary angles, complementary angles and right angles.angles, complementary angles and right angles.

G3.1.3: Find the image of a figure under the composition of two G3.1.3: Find the image of a figure under the composition of two or more isometries and determine whether the resulting or more isometries and determine whether the resulting figurefigureis a reflection, rotation, translation, or glide reflection is a reflection, rotation, translation, or glide reflection image of the original figure. image of the original figure.

L3.1.1: Distinguish between inductive and deductive reasoning, L3.1.1: Distinguish between inductive and deductive reasoning, identifying and providing examples of each.identifying and providing examples of each.

L3.1.2: Differentiate between statistical arguments (statements L3.1.2: Differentiate between statistical arguments (statements verified empirically using examples or data) and logical verified empirically using examples or data) and logical arguments based on the rules of logic. arguments based on the rules of logic.

L3.3.1: Know the basic structure for the proof of an “if…, then” L3.3.1: Know the basic structure for the proof of an “if…, then” statement and that proving the contrapositive is statement and that proving the contrapositive is equivalent.equivalent.


Inductive ReasoningInductive Reasoning

Inductive reasoning is based on Inductive reasoning is based on observations or patterns. observations or patterns.

Inductive reasoning is Inductive reasoning is NOTNOT valid valid for proofs.for proofs.


Inductive ReasoningInductive Reasoning

Betty observed 5 white cars Betty observed 5 white cars traveling very slowly down the traveling very slowly down the road so she concludes that all road so she concludes that all white cars are slow.white cars are slow.

Betty used inductive reasoning to Betty used inductive reasoning to reach her conclusion.reach her conclusion.


Deductive ReasoningDeductive Reasoning

Deductive reasoning is based on Deductive reasoning is based on known facts, or statements such known facts, or statements such as postulates definitions or as postulates definitions or theorems.theorems.

Deductive reasoning is valid for Deductive reasoning is valid for proofs.proofs.


Deductive ReasoningDeductive Reasoning

Triangle ABC is a right triangle so Billy Triangle ABC is a right triangle so Billy concludes triangle ABC has a right concludes triangle ABC has a right angle.angle.

Billy has used deductive reasoning Billy has used deductive reasoning because the definition of a right because the definition of a right triangle tells him that it has a right triangle tells him that it has a right angle.angle.

Which of the following statements Which of the following statements demonstrates deductive reasoning?demonstrates deductive reasoning?

A.A. All crows are black, and all crows are birds; All crows are black, and all crows are birds; therefore, all birds are black. therefore, all birds are black.

B.B. All dolphins have fins, and all fish have fins; All dolphins have fins, and all fish have fins; therefore, all dolphins are fish. therefore, all dolphins are fish.

C.C.Edward is a human being, and all human beings Edward is a human being, and all human beings are mortal; therefore, Edward is mortal. are mortal; therefore, Edward is mortal.

D.D.Megan gets good grades, and studying results in Megan gets good grades, and studying results in good grades; therefore, Megan is studying. good grades; therefore, Megan is studying.



Statistical Arguments vs Statistical Arguments vs Logical ArgumentsLogical Arguments

A statistical argument is made using _____ or A statistical argument is made using _____ or ___________ to justify your statements.___________ to justify your statements.

A logical argument is made by combining A logical argument is made by combining true statements (postulates, definitions true statements (postulates, definitions and theorems) together to reach a and theorems) together to reach a conclusion.conclusion.


Is this a statistical argument Is this a statistical argument or a logical argument?or a logical argument?

Barney read in his owners manual that he Barney read in his owners manual that he can increase his gas mileage by 15% if he can increase his gas mileage by 15% if he slows down by an average of 10 miles per slows down by an average of 10 miles per hour on the expressway. He then suggests hour on the expressway. He then suggests to his brother that he too should slow to his brother that he too should slow down to save gas. down to save gas.


Is Pebbles using a statistical Is Pebbles using a statistical argument or a logical argument or a logical

argument?argument?Given A, B and C are collinear points and Given A, B and C are collinear points and

that A and B are both on plane that A and B are both on plane QQ, Betty is , Betty is trying to determine if C must also be on trying to determine if C must also be on QQ. . Pebbles says C must be on Pebbles says C must be on QQ because of because of the Unique Plane Postulate.the Unique Plane Postulate.


Who is using a statistical Who is using a statistical argument?argument?

Harry picked 3 cards out of a deck of cards and selected 3 Harry picked 3 cards out of a deck of cards and selected 3 hearts.hearts.

Ron observed this and said the deck must not be a regular Ron observed this and said the deck must not be a regular deck of cards because there is no way you can draw 3 deck of cards because there is no way you can draw 3 straight hearts from a deck of cards.straight hearts from a deck of cards.

Hermione said it could be a regular deck because there is Hermione said it could be a regular deck because there is about a 1% chance of drawing 3 in a row of any suit.about a 1% chance of drawing 3 in a row of any suit.

A.A. only Rononly Ron

B.B. only Hermioneonly Hermione

C.C. only Harryonly Harry

D.D. Ron and HermioneRon and Hermione

E.E. no one is using a statistical argumentno one is using a statistical argument


Congruent Supplements Congruent Supplements Theorem:Theorem:

If 2 angles are supplements of congruent If 2 angles are supplements of congruent angles (or the same angle), then they are angles (or the same angle), then they are __________.__________.


Congruent Supplements Congruent Supplements Theorem:Theorem:

If ∠1 ≅ ∠ 3, ∠ 1 is supplementary ∠ 2 and ∠ 3 is supplementary ∠ 4, then ____________.

1 32 4


Vertical Angles Vertical Angles

Defn: Two angles are vertical angles iff they Defn: Two angles are vertical angles iff they are the nonadjacent angles formed by two are the nonadjacent angles formed by two intersecting lines.intersecting lines.

14

32


Vertical Angles Vertical Angles

14

32

Make a conjecture about vertical angles.

Try to justify your conjecture with mathematical statements.

Make a conjecture about vertical angles.

Try to justify your conjecture with mathematical statements.


Vertical Angle Theorem Vertical Angle Theorem


Prove the Vertical Angle Prove the Vertical Angle TheoremTheorem


Determine the measure of Determine the measure of ∠1∠1

1x2 + 2x +4 3

2x2 + 5x -50


Complete Activity 2 on page Complete Activity 2 on page 119.119.

You may work in pairs. You may work in pairs.

You have 10 minutes.You have 10 minutes.


Two Reflections Theorem for Two Reflections Theorem for Translations. Translations.

If a transformation is the composite of two If a transformation is the composite of two reflections over parallel lines, then it isreflections over parallel lines, then it is


You want to translate ΔABC 10 units by You want to translate ΔABC 10 units by reflecting it twice. Describe as accurately reflecting it twice. Describe as accurately as you can how to position the lines.as you can how to position the lines.


Two Reflections Theorem For Two Reflections Theorem For RotationsRotations

If a transformation is the composite of two If a transformation is the composite of two reflections over intersecting lines, then it reflections over intersecting lines, then it isis


Two Reflections Theorem for Two Reflections Theorem for RotationsRotations

F’

l

m70°

F

F”


Two Reflections Theorem for Two Reflections Theorem for RotationsRotations

l

m70°

F

F”

140°

Point P is reflected over line Point P is reflected over line mm and and then over line then over line nn. If the overall result . If the overall result is a rotation of 80 degrees, what is is a rotation of 80 degrees, what is the measure of the acute angle the measure of the acute angle formed by lines formed by lines mm and and nn??

A.A. 2020

B.B. 4040

C.C. 8080

D.D. 140140

E.E. 160160



AssignmentAssignment

pages 121 – 125, pages 121 – 125,

# 10 – 24 (evens), 25 – 27, 30, 32 – 34, 36, # 10 – 24 (evens), 25 – 27, 30, 32 – 34, 36, 38, 47, 48, 5038, 47, 48, 50

2.5: conjectures that lead to theorems

Documents

theorems expectations

right angles

right triangle

vertical angles

supplementary angles

complementary angles

following statements

true statements postulates