GEOMETRY UNITS 2 and 3

Logical Reasoning and Transformations

Name: _____________________________ Hour: _____________

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry Unit 2 and 3 Interactive Notebook

Geometry A Deductive System

Name _________________________

Hour __________

BC Cartoons by Johnny Hart and Creators Syndicate, Inc.

BC’s problem is trying to learn the meaning of “ecology” from Wiley’s Dictionary. This cartoon pokes fun at a problem every dictionary maker has. A dictionary attempts to give meaning to words by using simpler words or words the user may already understand. These words, in turn, are defined by even simpler words, but the process cannot go backward without an end. Every dictionary solves this problem by going around in circles. For the same reason that it is impossible to define everything without going in circles, it is impossible to prove everything.

Adapted from Harold R. Jacobs. Geometry: Seeing, Doing, Understanding, 3rd Edition. New York: Freeman, 2003

You already know a great many geometric facts. It seems clear that we should try to organize our knowledge of geometry. We shall state definitions as clearly and precisely as we can; and we shall prove various geometric facts by means of logical reasoning. The statements that we prove are called theorems. We would like to prove each theorem by showing that it follows

logically from theorems that have already been proved. But this is impossible. The first theorem cannot be proved that way because there are no previously proved theorems. We have to start somewhere, and so we must accept some statements without proof. These unproved statements will be our postulates or axioms. Just as we must start with some unproved statements, so must we start with some undefined terms which will be point, line and plane.

Adapted from Anderson, Garon, Gremillion. School Mathematics Geometry

Geometry Rube Goldberg Inventions

Name _________________________

Hour __________

Rueben Garret Lucius Goldberg was born on July 4, 1883 in San Francisco. By the end of his life (December 7, 1970), he was known as an inventor, sculptor, author, engineer and cartoonist and would make a mark in history for his extraordinary achievements. His cartoon strips were popular but the work that gave him unforgettable lifelong fame was the character he created, Professor Lucifer Gorgonzola Butts. Using the character, he would illustrate inventions that later become known as “Rube Goldberg Machines”. A “Rube Goldberg Machine” is an extremely complicated device that executes a very simple task in a complex, indirect way. This is now used as an expression to describe any system that's confusing or complicated and came about from Goldberg's illustrations of absurd machines.

Adapted from http://www.rube-goldberg.com/

1) Here is one of Rube Goldberg’s cartoons. It is titled “Automatic arrangement which

reminds you to get a haircut.”

Complete the implications of this machine. A →B Long hair (A) tickles the chin of the miniature mule (B).

B →C Mule (B) kicks causing spoon (C) to squirt grapefruit.

C →D Squirt grapefruit (C) into eye of guest (D).

2) How to Stay Awake in Class

Designed by Jennifer Doyle, student at Oakland Technical High School

Write the implications for this situation.

3) Create a “Rube Goldberg” invention that will turn on your teacher’s computer in a classroom when you open the classroom door. Your invention must have at least five (5) implications.

Geometry Reasoning

Name _________________________

Hour __________

Read the three definitions for each term. Write your own definition and give two examples for each term, including one non-mathematical and one mathematical example. Definitions:

Inductive Reasoning: Method used for drawing conclusions from a limited set of

observations. 1

It is the process of observing data, recognizing patterns, and making generalizations about those patterns. When you use inductive reasoning to make a generalization, the generalization is called a conjecture.2

Reasoning based on patterns you observe.3

Deductive Reasoning: Uses logic to draw conclusions from statements already

accepted to be true.1

The process of showing that certain statements follow logically from agreed upon assumptions and proven facts.2

The process of reasoning logically from given statements to a conclusion.3

1 Jacobs, Harold R. Geometry: Seeing, Doing, Understanding, 3

rd Edition. New York: Freeman, 2003

2 Serra, Michael. Discovering Geometry: An Investigative Approach, 3

rd Edition. Emeryville, CA. Key Curriculum, 2003

3Bass, Charles, Hall, Johnson, Kennedy. Geometry. Boston, MA. Prentice Hall, 2007

b

a

Geometry Venn Diagrams

Name _________________________

Hour __________

In Agatha Christie’s “Murder on the Orient Express” a murder has occurred. One scenario is that the conductor, who left the train because the train is stuck in a snow bank, is the murderer. The detective is logically trying to solve the murder. He uses a crushed watch to determine the time of death. So the detective says “If the crime was committed at a quarter past one, then the murderer could not have left the train.” The detective’s statement is an example of a conditional statement, where the “if” part is the hypothesis and the “then” part is the conclusion. A conditional statement can be represented symbolically by a → b read as “If a, then b.”

In the Murder on the Orient Express, a represents the words “the crime was committed at a quarter past one” and b represents the words “the murderer could not have left the train.” This situation can also be modeled using Venn diagrams where the inner circle represents the “a” part of the conditional statement, and the outer circle represents the “b” part of the conditional statement.

you are in

Michigan

you are

crossing

the

Mackinac

Bridge

1) If a flag is the current United States Flag, then it has 50 stars on it.

a) Create a Venn diagram for this situation. b) What is the hypothesis of this conditional situation? c) What is the conclusion of this conditional situation?

2) a) Write a conditional statement

for this situation.

b) What is the hypothesis of this conditional situation?

c) What is the conclusion of this conditional situation?

3) Two angles whose sum is 1800 are supplementary.

a) Write a conditional statement for this situation.

b) Create a Venn diagram for your conditional statement from part a.

c) What is the hypothesis of this conditional situation?

d) What is the conclusion of this conditional situation? 4) Given this statement: All Detroit Red Wings players make a lot of money. Which of

the following statements are supported by this statement? Explain your answer. a) If you are a Detroit Red Wing player, then you make a lot of money.

b) If you make a lot of money, then you are a Detroit Red Wing player.

5) A right angle measures 900. Which of the following statements are supported by this

statement? Explain your answer. a) If an angle is a right angle, then it measures 900.

b) If an angle measures 900, then it is a right angle.

1+3

2+4

5

4

3

21

Geometry What are they proving?

Name _________________________

Hour __________

This proof appeared in a book titled Proof without Words: Exercise in Visual Thinking.

1. What do you think is being proved?

2. Briefly explain in words how the proof works.

Geometry Proving Constructions

Name _________________________

Hour __________ First complete the construction and then prove why the construction is true.

1. Perpendicular bisector of AB

A B

2. Angle bisector of ABC

B

A

C

3. Construct a line perpendicular to a given line through a point not on the line

Geometry Proof Involving Quadrilaterals

Name _________________________

Hour __________

1. a. Set up the coordinates for the four vertices of the given parallelogram.

b. Prove that your coordinates constructed a parallelogram.

c. Define the term segment bisector.

d. Write the definition of segment bisector as a biconditional, iff, statement.

e. Prove that the diagonals of a parallelogram bisect each other.

2) a. Set up a general square on a coordinate system.

b. Prove that your coordinates constructed a square.

c. Define the term congruent segments.

d. Write the definition of congruent segments as a biconditional, iff, statement.

e. Prove that the diagonals of a square are congruent.

f. Define the word perpendicular.

g. Write the definition of perpendicular as a biconditional, iff, statement.

h. Prove that the diagonals of a square are perpendicular.

3) a. Setup a general rectangle on a coordinate system. b. Prove that your coordinates

constructed a rectangle. c. Prove that the diagonals of a rectangle are congruent. d. Prove that the diagonals of a rectangle do not have to be perpendicular.

4) a. Setup a general rhombus on a coordinate system.

b. Prove that your coordinates constructed a rhombus.

c. Prove that the diagonals of a rhombus are perpendicular.

d. Prove that the diagonals of a rhombus do not have to be congruent.

5) a. Setup a general isosceles trapezoid on a coordinate system.

b. Prove that your coordinates constructed an isosceles trapezoid.

c. Prove that the diagonals of an isosceles trapezoid are congruent.

Geometry Proving an Equilateral Triangle

Name_________________________ Hour______________

Given a Square piece of paper of length x Prove the ending triangle is equilateral

Adapted from Macomb Mathematics Science Technology Center

-30 -20 -10 10 20 30

20

15

10

5

-5

-10

-15

-20

x=2x=-3

x

y

-10

-5

C

B

A

Geometry Composite of Reflections over Two Parallel Lines

Name _________________________

Hour __________

1. Using a colored pencil, reflect ABC over the x = -3 line and label the points A', B', and C' respectively. Draw A'B'C'.

2. Using a black pencil, reflect A'B'C' over the x = 2 line and label the points A", B", and C" respectively. Draw A"B"C".

3. Draw arrows from A to A", from B to B", from C to C" using a different color. 4. What transformation occurred that would map ABC onto A"B"C"?___________

5. How far did ABC move to become A"B"C"?___________in what

direction?_______

6. Write a composite for this situation that maps the first triangle to the last triangle.

-30 -20 -10 10 20 30

20

15

10

5

-5

-10

-15

-20

x=2x=-3

x

y

-10

-5

C

B

A

-20 -10 10 20 30

15

10

5

-5

-10

-15

-20

-5

5

-10

y

x

x = -2x = -7

C

B

A

7. Using a colored pencil, reflect ABC over the x = 2 line and label the points A', B', and C' respectively. Draw A'B'C'.

8. Using a black pencil, reflect A'B'C' over the x = -3 line and label the points A", B", and C" respectively. Draw A"B"C".

9. Draw arrows from A to A", from B to B", from C to C" using a different color. 10. What transformation occurred from ABC to become A"B"C"?___________

11. How far did ABC move to become A"B"C"?___________in what direction?_______

12. Write a composite for this situation that maps the first triangle to the last triangle.

13. Using a colored pencil, reflect ABC over the x = -7 line and label the points A', B', and C' respectively. Draw A'B'C'.

14. Using a black pencil, reflect A’B’C’ over the x = -2 line and label the points A", B", and C" respectively. Draw A"B"C".

15. Draw arrows from A to A”, from B to B”, from C to C” using a different color. 16. What transformation occurred from ABC to become A"B"C"?___________

17. How far did ABC move to become A"B"C"?___________in what direction?_______

18. Write a composite for this situation that maps the first triangle to the last triangle.

-30 -20 -10 10 20 30

20

15

10

5

-5

-10

-15

-20

x

y

5

-5

C

B

A

19. Conjecture on any relationship there might be between the distance between the lines and the distance the original triangle moves in relationship to the ending triangle.

20. Graph the line x = 4. Find a second line of reflection so that the composite of the two reflections will translate ABC 10 units to the right. Write the composite.

-30 -20 -10 10 20 30

20

15

10

5

-5

-10

-15

-20

x

C

B

A

21. Graph the lines y = 3 and y = -2. 22. Using a colored pencil, reflect ABC over the y = 3 line and label the points A', B',

and C 'respectively. Draw A'B'C'. 23. Using a black pencil, reflect A’B’C’ over the y = -2 line and label the points A", B",

and C" respectively. Draw A"B"C". 24. Draw arrows from A to A", from B to B", from C to C" using a different color. 25. What transformation occurred from ABC to become A"B"C"?___________

26. How far did ABC move to become A"B"C"?___________in what

direction?_______

27. Write a composite for this situation that maps the first triangle to the last triangle.

-30 -20 -10 10 20 30

20

15

10

5

-5

-10

-15

-20

x

C

B

A

28. Graph the lines y = 3 and y = -2. 29. Using a colored pencil, reflect ABC over the y = -2 line and label the points A', B',

and C' respectively. Draw A’B’C’. 30. Using a black pencil, reflect A'B'C' over the y = 3 line and label the points A"B"C"

respectively. Draw A"B"C". 31. Draw arrows from A to A", from B to B", from C to C" using a different color. 32. What transformation occurred from ABC to become A"B"C"?___________

33. How far did ABC move to become A"B"C"___________in what

direction?_______

34. Write a composite for this situation that maps the first triangle to the last triangle. 35. What conjectures can you make about the composite of two reflections over two parallel lines?

Adapted from Macomb Mathematics Science Technology Center

Geometry Congruent Triangles

Name _________________________

Hour __________ Problem: Is it possible to construct congruent triangles if we only know three parts of the triangle? 1. Given ΔABC with the length of AB, BC and the measure of B

A B

B C

B

2. Given ΔDEF with the length of DE, and the measures of D and E

D E

D

E

3. Here is ΔGHK. Are there any other combinations of 3 parts that can produce a unique triangle congruent to ΔGHK? What are they? Construct your solutions.

H

GK

A

B

CD

Geometry Dilation

Name _________________________

Hour __________

Using a compass and a straight edge:

1. Dilate ∆BCD by a factor of 3 from center A

A

B

CD

2. Dilate ∆BCD by a factor of -2.5 from center A

an angle is less than 90 degrees

it is an acute angle

an angle is bisected

a ray divides the angle into two congruent angles

two angles have a common side and their other sides are opposite rays

the angles form a linear pair

two legs of a trapezoid are congruent

the trapezoid is an isosceles trapezoid

two angles form a linear pair

the angles are supplementary

two lines form a right angle

the lines are perpendicular

four sides of a quadrilateral are congruent

the quadrilateral is a square

the four angles of a quadrilateral are congruent

the quadrilateral is a rectangle

the alternate interior angles are congruent

the lines are parallel

two triangles are congruent

two triangles have the same area

two angles are vertical angles

the two angles have equal measure

the quadrilateral is a parallelogram

the quadrilateral is a rhombus

B

C

A

x

y

10

105

5

A'

C'

B'

A'

C'

B'

5

5 10

10

y

x

A

C

B

Geometry Composite of Reflections over Two Intersecting Lines

Name _________________________

Hour __________ 1 – 4 Find the line of reflection and highlight it with a colored pencil. Write the equation of the line of reflection. Find the coordinates of the reflected image and use them to write the image formula that would reflect any point (x,y) over the given reflection line. 1.

2.

A (2,2) B (5,8) C (8,5)

A' B' C'

Equation of the line of reflection

Image formula for this reflection

A (2,2) B (5,8) C (8,5)

A' B' C'

Equation of the line of reflection

Image formula for this reflection

C'

A'

B'

5

5 10

10

y

x

A

C

B

C

B

A

x

y

10

105

5

C'

B'

A'

5

5 10

10

y

x

A

C

B

3.

4.

5 – 6 Use the image formulas written in numbers 1 – 4 to find the new coordinates of ABC. Then graph the new triangle (A'B'C').

A (4,-7) B (2,-3) C (9,-1)

A' B' C'

Equation of the line of reflection

Image formula for this reflection

A (2,2) B (5,8) C (8,5)

A' B' C'

Equation of the line of reflection

Image formula for this reflection

5. Equation of the line of reflection: y = x Image formula for this reflection: _________

A (-6,2) B (-3,9) C (3,7)

A' B' C'

C

B

A

x

y

10

105

5

C

B

A x

y

10

105

5

7 – 8 Use the image formulas written in numbers 1 – 4 to first find the coordinates of A'B'C' and then to find the coordinates of A"B"C" for the given composites. Then graph ONLY A"B"C". Also graph the two lines of reflection.

6. Equation of the line of reflection: y = -x Image formula for this reflection: _________

A (-6,2) B (-3,9) C (3,7)

A' B' C'

7. Composite: (ry=x ◦ ry=0 )( ABC)

Equation of the first line of reflection: y = 0 Equation of the second line of reflection: y = x

A (3,-2) B (1,-6) C (6,-10)

A' B' C'

A" B" C"

What transformation occurred from this composite? In other words, what transformation would transform ABC to A"B"C" without using any reflections?

Give an image formula for this transformation.

C

B

A x

y

10

105

5

9. Notice in numbers 7 and 8, the same two lines of reflection were used, however the image triangle is located in a different position. Make a conjecture on what you think makes the difference.

8 Composite: ry=0(ry=x(ABC))

Equation of the first line of reflection: y = x Equation of the second line of reflection: y = 0

A (3,-2) B (1,-6) C (6,-10)

A' B' C'

A" B" C"

What transformation occurred from this composite? In other words, what transformation would transform ABC to A"B"C" without using any reflections?

Give an image formula for this transformation.

C

B

A x

y

10

105

5

10 – 11 Use the image formulas written in numbers 1 – 4 to first find the coordinates of A'B'C' and then to find the coordinates of A"B"C" for the given composites. Then graph ONLY A"B"C".

10. Composite: (ry=x ◦ ry=-x)( ABC)

Equation of the first line of reflection: ______ Equation of the second line of reflection: _____

A (3,-2) B (1,-6) C (6,-10)

A' B' C'

A" B" C"

What transformation occurred from this composite? In other words, what transformation would transform ABC to A"B"C" without using any reflections?

Give an image formula for this transformation.

C

B

A x

y

10

105

5

12

12. Notice in numbers 10 and 11, the same transformation occurred but different lines were used. Make a conjecture on what you think must be true about the lines used as lines of reflection.

11 Composite: rx=0(ry=0(ABC))

Equation of the first line of reflection: ______ Equation of the second line of reflection: _____

A (3,-2) B (1,-6) C (6,-10)

A' B' C'

A" B" C"

What transformation occurred from this composite? In other words, what transformation would transform ABC to A"B"C" without using any reflections?

Give an image formula for this transformation.

C

B

A x

y

10

105

5

C

B

A x

y

10

105

5

Extension:

A. Find two lines of reflection that can be used to write a composite forABC that will produce a 900 rotation ofABC with center (3,3) and write their equations.

B. Find two lines of reflection that can be used to write a composite forABC that will produce a -900 rotation ofABC with center (2,-1) and write their equations.

Geometry Proof of Composite of Reflections over Two Parallel Lines

Name _________________________

Hour __________

Prove that the composite of two reflections over two parallel lines produces a translation that is double the distance between the lines and in the direction of the mapping of the first line to the second line. Case One – If the point is between the two parallel lines.

Proof: 1. Construct a line perpendicular to lines 1 and 2. Since the lines are parallel a line perpendicular to one of the lines would be perpendicular to the other parallel line. Label the intersection with line 1 as point B and the intersection with line 2 as point C. Label the distance from A to B as x; AB = x

Line 1 Line 2

A

B x A C

Line 2 Line 1

Given: Line 1 ║ Line 2 rLine 2 ◦ r Line1 ( point A)

Find A’ after A has been reflected over Line 1. Mark A’B = x because reflection maintains distance.

Reflect A’ over line 2 and label A”

Since the distance from A to C is unknown, label that distance y; AC = y. Therefore; the distance from C to A” is 2x + y; A”C = 2x + y.

A' x B x A C

Line 2 Line 1

A' x B x A C A''

Line 2 Line 1

The distance between the two parallel lines is BC = x + y The distance between A and A” is AA” = 2x + 2y Therefore, the distance that A translated because of the composite from A to A” is twice the distance between the two lines and the translation moved right because Line 1 moves right to map to Line 2. You are to prove the other four cases!

A' x B x A y C 2x + y A''

Line 2 Line 1

Case Two – The point is on the left of the two parallel lines.

Case 3 – The point is on the right of the two parallel lines.

A

Line 1 Line 2

A

Line 2 Line 1

Given: Line 1 ║ Line 2

rLine 2 ◦ r Line1 ( point A)

Given: Line 1 ║ Line 2

rLine 2 ◦ r Line1 ( point A)

Cases four and five. If the point is on one of the reflecting lines.

Line 2

Line 1

A

Geometry Proof of Composite of Reflections over Two Intersecting Lines

Name _________________________

Hour __________ The composite of two reflections over two intersecting lines produces a rotation centered at the intersection of the lines that is double the rotation of the first line to the second line. There are five parts to this proof. Part One

rLine 2 ∘ r Line 1 (point A)

The composite of two reflections over two intersecting lines produces a rotation centered at the intersection of the lines that is double the rotation of the first line to the second line. There are five parts to this proof. Part Two

Line 2

Line 1

A

rLine 2 ∘ r Line 1 (point A)

The composite of two reflections over two intersecting lines produces a rotation centered at the intersection of the lines that is double the rotation of the first line to the second line. There are five parts to this proof. Part Three

Line 2

Line 1

A

rLine 2 ∘ r Line 1 (point A)