Modern Geometry Math 109

 

 

 

Section 1.1

 

Statements and Reasoning

 

Statement:  A statement is a group of words and symbols that can be classified as true or false.

 

Examples of statements

 

Geometry is a mathematics course

 

Thomas Jefferson was from Virginia

 

Example of something that is not a statement

 

Help me!

 

Compound statement

 

Two or more statements that form a new statement

 

Types of Compound Statements

 

Let P and Q be statements

 

Conjunction:  P and Q

 

Disjunction:  P or Q

 

Negation; ~P

 

Examples

 

The sky is blue the sea is green   (Conjunction)

 

John will either take Geometry or Algebra (Disjunction)

 

Let P be the statement “John likes to take math classes”, then the negation of P (~P) would be “John doesn’t like to take math classes”

 

 

 

 

 

Types of Reasoning

 

1) Intuition:   An inspiration leading to the statement of a theory.

 

2) Induction:    An organized effort to test the theory.

“Start from specifics and derive a general rule”.

 

3) Deduction:  A formal argument that proves the tested theory

 

Examples

 

Mary walks into her Chemistry class and says I think Professor Brown is going to give us a test today.   (Intuition)

 

While driving through Virginia on the interstate you notice that the first 6 information signs you pass are blue, so you conclude that all information signs in Virginia are blue.

(Induction)

 

You notice that a quadrilateral has four congruent sides and four right angles, so you conclude that the quadrilateral is a square.  (Deduction)

 

If -Then statements  (Conditional)

 

If P then Q

 

P is the hypothesis

 

Q is the conclusion

 

Example

 

If it rains tomorrow, then I will bring an umbrella to work.

 

Hypothesis – If it rains tomorrow

 

Conclusion – I will bring an umbrella to work

 

 

 

 

 

 

 

 

Examples from the homework set 1.1 starting on page 7

 

2) Determine if the following are statements

 

a) Chicago is located in the state of Illinois.  (Statement)

 

b) Get out of here! (Not a statement) 

 

c)  x<6, when x = 10  (Statement)

 

d) x+3 = 7 when x = 5  (Statement)

 

 

 

4) Find the negation (~P) of each statement P

 

a)  No one likes me = P

~P = Everyone likes me

 

b) P = Angle 1 is a right angles

~P = Angle one is not a right angle

 

Types of Compound Statements

 

Conditional “If, then’

Disjunction “or’

Conjunction “and”

 

Page 7

 

Classify each statement as a compound statement or simple statement.  If the statement is compound tell if is a conditional, conjunction, or disjunction.

 

6) Alice played and the team won   (Compound  Statement, Conjunction)

 

8) An integer is odd or it is even  (Compound Statement, Disjunction)

 

10) You will be in trouble if you don’t change you ways.  (Compound, Conditional)

 

 

 

 

 

 

 

 

Identify the hypothesis and conclusion

 

14) If   , where , then

 

Hypothesis:    If  , where

 

Conclusion:    

 

 

16) If two lines intersect, then the vertical angles are congruent.

 

Hypothesis: If two lines intersect

 

Conclusion:  the angles congruent

 

18) If the angle is a base angle of an isosceles triangle, the angles are congruent.

 

Hypothesis: If the angle is a base angle of an isosceles triangle

 

Conclusion: the angles are congruent

 

 

 

Classify each statement as true or false

 

20) Rain is wet and snow is cold.

 

True

 

22) If Jim lives in Idaho, then he lives in Boise.

 

False

 

24) Triangles are congruent or circles are round.

 

True

 

 

 

 

 

 

 

 

Identify each type of reasoning as Induction, Deduction, or Intuition

 

26) You walk into your geometry class, look at the teacher, and conclude that you will have a quiz today.   (Intuition)

 

30) While judging at science fair project, Mr. Lange finds that the each of the first 5 projects is outstanding, and concludes that all 10 projects will be outstanding.              (Induction)

 

 

 

Section 1.2

 

Line and Angle Relationships

 

Undefined Geometric Terms

 

A point, line, ray

 

Examples

 

 

 

                            

 

 

 


 

 

 

 

 

 

Defined Terms

 

Collinear:  Three or more points that lie on the same line.

 

Non-Collinear; Three or more point that do not lie on the same line

 

Angle: The union of two rays that meet at a common endpoint called the vertex.

 

An angle with vertex A

 

 

 

 

Representations of rays, lines, and segments

 

Object

          Drawing

Representation

Point

 

            

A

 

 

Line Segment

 

 

 

 

     

             

 

 

 

 

Line

 

 

                                                                                         

                                                                                                       

 

 

 

 

Ray

 

 

 

 

 

 

 

 

 

 

 

Angle

 

 

 

 

         

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Exercises Section 1.2  page 16

 

4) Do the following points appear to be collinear

 

 

The points appear to noncollinear

 

6) How many lines are formed from points A, B, and C

 

 

 

 

Three lines are formed by A,B, and C

 

 

Definition:  An angle’s measure must be between 0 degrees and 180 degrees.

 

8)  Determine if the following measurements can be angle measures

 

 

 

 

 

10) Draw A-X-B (Segment AB with X lying between A and B)

 

 

 

 

12) List the all ways to name

 

 

 

,,and

 

14)  Compare angles 1 and 2

 

                          

                                   

 

 

 

 and the point of intersection is R

 

34)

       

 

 

36) Use drawing form #34

 

 

38)  

 

  

 

Drawing for 40 and 42 

 

 

 

 

 

 

 

 

 

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Section 1.3

 

Mathematical systems and Postulates

 

Mathematical Systems

 

1)      Undefined Terms

2)      Defined Terms

3)      Axioms and Postulates

4)      Theorems

 

Definition:   An isosceles triangle that has two congruent sides

 

Definition:  A line segment is the part of a line that consists of two points, known as endpoints and all points between them.

 

Euclidean Set of Postulates

 

Postulate 1:  Through two distinct points, there is exactly one line.

 

Postulate 2:  The measure of any line segment is a unique positive number.

 

Definition:   The distance between two points A and B is the length of the line segment that joins the two points.

 

Postulate 3: If x is a point of and A-X-B, then

(Segment Addition Postulate)

 

Postulate 4:  If lines intersect, they intersect at a point.

 

 

 

 

 

 

Definition:  Parallel lines are lines that do not intersect and do not lie in the same plane.

 

 

Postulate 5:  Through three non-collinear points there is exactly one plane

 

 

 

 

Definition: The midpoint of a line segment is a point that is equal distant from both endpoints

 

Postulate 6:  If two distinct points intersect, they intersect to form a line.

 

 

 

 

 

 

 

 

 

 

 

 

 


Postulate 7:  Given two distinct points in a plane, the line containing these points also lies in the plane.

 

 

Exercises Section 1.3

 

2)

 

 

If, then B is the midpoint of  

 

8)         

 

      

                                

How many lines are determined by given points

a)      Point A (Infinite Lines)

b)      Points A and B  (One Line)

c)      Points A,B,C (None)

d)      Points A,B,D (One Line)

 

12)

 

                              

Determine if the following statements are true or false.

 

a)

b)

c)

d)

e)

 

14)  

             

 

 

 

 

16) Can a segment bisect a line?    No

 

       Can a line bisect a segment?    Yes

 

18)  Points C and D lie in plane X

                                                            

 

 

 

 

 

 

20) Sketch intersecting planes

 

 

 

 

 

 

 

 

 

 

 


Sketch parallel planes

 

                                                    M

 

 


                                                  N

 

 

 

 

                                  

24) 

 

                                        

       

a)      Points A,B, and C lie in the same plane.

b)      Points B,C, and D form infinitely many planes.

c)      Points A,B,C, and D share one plane.

d)      Points A,B,C, and E are non-coplanar.

 

 

 

 

 

 

Example            

 

         

               A                    B

           D                C    

 

 

 

 

                                     F

 

             E               C

 

1) Are the points A, B, and D coplanar?  Yes

 

2) Are the points A, B, and E coplanar?  Yes

 

3) Are the points A, B, and E collinear?  No

 

4) Are the points A, D, C, and F coplanar?  Yes

 

5) Are the points E, F, D, and A coplanar?  No

 

 

 

 

 

 

 

 

 

40) 

 

 

 

42)

 

 

 

 

 

 

 

 

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 ­­­­­­­­­­­­­Section 1.4

 

 

Angle Relationships

 

Definition:   An angle is the union of two rays that share a common endpoint.

 

Postulate 8

 

The measure of an angle is a unique positive number.

 

Types of Angles

 

An acute angle is an angle whose angle measure is greater than and less than  

 

An obtuse angle is an angle is whose measure is greater than and less than  

 

A right angle is an angle with a measure of

 

 

Postulate 9:  Angle Addition Postulate

 

If a point D lies in the interior of angle ABC, then

 

 

Complementary Angles:  Two Angles whose angle measure sum is

 

Supplementary Angles:  Two Angles whose angle measure sum is

 

Adjacent Angles: Two angle who share a common side and common vertex, but have no interior points in common.

 

 

 

 

Vertical Angles:  Two nonadjacent angles that form from intersecting lines.

 

 

 

 

 

Exercises

 

2) Describe each angle

 

a)  

 

b)

 

c)

 

6)

 

 

 

 

 

8)

          

are supplementary angles and adjacent angles

 

 

10)                                   

 

 

a)

 

b)

 

c)

 

e) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14)

 

 

 

 

 

 

 

 

16) 

 

 

 

18)

 

 

 

 

 

 

 

 

 

 

 

 

20)

 

 

Since ST bisects,

 

Thus, 

 

Subtract 4 from the second equation and get

 

 

Multiply this equation by 3

 

 

Add to the first equation will eliminate the y variable

 

 

 

 

Math 109 Section 1.4

In class Assignment

Use the figure below to answer Problems 1 and 2

 

 

1)

 

 

 

2)

 

 

3) Solve for x and y.

 

 

4) Solve for x and y

 

 

 

 

 

 

 

 

 

 

 

Solutions

 

1)

 

2)

 

3)

 

 

 

4)

 

 

 

 

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Section 1.5

 

Introduction to Proofs

 

Algebraic Proofs

 

Properties of Equality

 

Addition Property of Equality

 

 

Subtraction Property of Equality

 

 

Multiplication Property of Equality

 

 

Division Property of Equality

 

 

Distributive Property

 

 

Transitive Property of Equality

 

 

Substitution Property

 

 

 

 

 

 

 

 

 

Algebraic Proofs

 

1) Given:

     Prove: x = 2

 

 

Statement

 

 

Reason

 

1)

1) Given

 

2)

2) Subtraction Property of Equality

 

3)

3) Substitution Property

 

4)

4) Division of Equality

5)

5) Substitution Property

 

 

 

 

2)  Given:

     Prove: x = 2

 

 

Statement

 

 

Reason

 

1)

1) Given

 

2)

2) Distributive Property

 

3)

3) Subtraction Property of Equality

 

4)

4) Substitution property

 

5)

5) Division Property of Equality

 

6)

6) Substitution Property

 

 

 

 

 

 

 

3) Given:

     Prove: x = 14

 

 

Statement

 

 

Reason

 

1)

1) Given

 

2)

2) Subtraction Property of Equality

 

3)

3) Substitution Property

 

4)

4) Multiplication of Equality

5)

5) Substitution Property

 

 

Problems from page 43

 

Name the appropriate property

 

2)

 

Substitution Property

 

4)

 

Addition Property of Equality

 

6)

 

Addition Property of Equality

 

Use the information given to draw a conclusion based on the state property or definition.

 

12) Given M is the midpoint of

 

Use the definition of a midpoint to make a conclusion

 

 

 

14) Given:

Use the definition of an angle bisector

 

 

   

Conclusion:

 

 

 

16) Use the above drawing                                                                                                 

 

 

Use the definition of complementary angles

 

are complementary angles

 

 

 

 

Geometric Proofs

 

 

 

 

 

 

Page 44

 

#22

 

Given:    E is the midpoint of

 

Prove:    

 

 

 

 

 

Statement

 

 

Reason

 

1) E is the midpoint of

 

1) Given

2)

2) Definition of a midpoint

 

3)

3) Segment Addition Property

 

4)

4) Substitution Property of Equality

 

5)

5) Substitution Property

 

6)

6) Division Property of Equality

 

 

 

 

 

 

 

 

 

Math 109 Section 1.4

Use the figure below to answer Problems 1 and 2

 

 

1)

 

 

 

2)

 

 

3) Solve for x and y.

 

 

4) Solve for x and y

 

 

 

 

 

 

 

 

 

 

 

Solutions

 

1)

 

2)

 

3)

 

 

 

4)