page 53, chapter summary: concepts and procedures after studying this chapter, you should be able...

, Chapter Summary: Concepts and Procedures

After studying this CHAPTER, you should be able to . . .

1.1 Recognize points, lines, segments, rays, angles, and triangles.1.2 Measure segments and angles1.2 Classify angles and name the parts of a degree1.2 Recognize congruent angles and segments1.3 Recognize collinear and noncollinear points1.3 Recognize when a point is between two other points1.3 Apply the triangle inequality principle1.3 Correctly interpret geometric diagrams1.4 Write simple two-column proofs1.5 Identify bisectors and trisectors of segments and angles1.6 Write paragraph proofs1.7 Recognize that geometry is based on a deductive structure1.7 Identify undefined terms, postulates, and definitions1.7 Understand the characteristics and application of theorems1.8 Recognize conditional statements and the negation, the converse, the inverse, and the contrapositive of a statement1.8 Use the chain-rule to draw conclusions1.9 Solve probability problems 2

Chapter 1, Section 1: “Getting Started”

1.1 Recognize points, lines, segments, rays, angles, and triangles.

ANGLE

INTERSECTION

LINE LINE SEGMENT

NUMBER LINE

POINT

RAYSEGMENT

TRIANGLE

UNION

VERTEX

After studying this SECTION, you should be able to . . .

Related Vocabulary

ENDPOINTS

3


ANGLE

LINE LINE SEGMENT

POINT

SEGMENT


Related Vocabulary

ENDPOINTS


4


INTERSECTION

NUMBER LINE

RAY TRIANGLE

UNIONVERTEX


Related Vocabulary

0 1 2 3- 3

- 2 - 1


5

Your Turn!

What’s My Name?

1.

Q

2.

3.

4.

6

A TC

GOD

EB

point Q or Q

ray CA or ray CT

CA CT

line DG …or line DO, GD, GO, or OD

GOGDDODG

segment BE or segment EB

EBBE

To see answers, hit space bar.

OD

Easy peasy!


1.2 Measure segments and angles

1.2 Classify angles and name the parts of a degree

1.2 Recognize congruent angles and segments

ACUTE ANGLECONGRUENT ANGLES

CONGRUENT SEGMENTS

MEASURE

MINUTES

OBTUSE ANGLE

PROTRACTOR

RIGHT ANGLE

SECONDS

DEGREES

STRAIGHT ANGLE

TICK MARK

Related Vocabulary

Chapter 1, Section 2: “Measurement of Segments and Angles”

7





ACUTE ANGLE

OBTUSE ANGLE

RIGHT ANGLE STRAIGHT ANGLE

Related Vocabulary


8





CONGRUENT ANGLES

CONGRUENT SEGMENTS

Related Vocabulary


9





MEASURE

Degrees & MINUTES

PROTRACTOR RULER

Degrees, Minutes, & SECONDS

DEGREES

TICK MARK

Related Vocabulary


360⁰ 359⁰ 60’ 359⁰ 59’ 60”

10

1.3 Recognize collinear and noncollinear points

1.3 Recognize when a point is between two other points

1.3 Apply the triangle inequality principle

1.3 Correctly interpret geometric diagrams

COLLINEAR NONCOLLINEAR

Related Vocabulary


Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”

X Y Z X Z

Y

X, Y, and Z are collinear

X, Y, and Z are NOT collinear

11





COLLINEARITY Betweenness of Points

Related Vocabulary



X Y Z

Y is between X and Z

X Z

Y

Y is NOT between X and Z

12





POSTULATE:



The sum of the measures of any two sides of a triangle is always greater than the measure of the third side.

Nope!Nope!YES!

13





TRIANGLE INEQUALITY:



For any three points, there are only two possibilities:

YES!

1. They are collinear (one point is between the other two,

such that two of the measures equals the third, or

2. They are noncollinear (the three points determine a triangle!

The sum of any two sides exceeds the measure of the third side! 14





See very important TABLE on page 19!



Do Assume:AD and BE are straight lines

∡BCE is a straight angleC, D, and E are

noncollinearC is between B and E

E is to the right of A

AC

D

E

B

Allowable Assumptions:• Straight lines• Straight angles• Noncollinearity• Betweenness of points• Relative position of points

15





See very important TABLE on page 19!



DO NOT Assume:

AB ≅ CD

∡BAC is a right angle

∡B ≅ ∡E∡CDE is an obtuse angle BC is longer than CE

AC

D

E

B

Forbidden Assumptions:• Right Angles• Congruent segments• Congruent Angles• Relative SIZES of segments• Relative SIZES of angles

16

You must PROVE these!

1.4 Write simple two-column proofs


Related Vocabulary

THEOREM

TWO-COLUMN PROOF

Chapter 1, Section 4: “Beginning Proofs”

- a mathematical statement that can be proved

#4 Statements

#5 Reasons

- A step-by-step logical argument offering proof by a chain of statements and reasons in support of a specific conclusion. A two-column proof has FIVE

parts:1. Givens 2. Prove 3. Diagram

Example, Theorem 1: If two angles are right angles, then they are congruent.

#1 Given: ∡A is a right ∡ ∡B is a right ∡#2 Prove ∡A ≅ ∡B

#3 Diagram

A

B

1. ∡A is a right ∡3. ∡B is a right ∡2. m∡A = 904. m∡B = 905. ∡A ≅ ∡B

1. Given

2. If an ∡ is a right ∡, then its measure is 90

3. Given

4. Same as #25. If 2 ∡’s have the same

measure, then they are ≅17

1.5 Identify bisectors and trisectors of segments and angles

Related Vocabulary


BISECT BISECTOR

MIDPOINT

TRISECT TRISECTORS

TRISECTION POINTS

Chapter 1, Section 5: “Division of Segments and Angles”

18


Related Vocabulary


BISECT BISECTOR

MIDPOINT


(verb) to divide into two congruent parts

(noun) the POINT that divides a segment into two congruent segments

(noun) the name of the point that divides a segment into two congruent segments

Question: Is it possible for a line to have a MIDPOINT?

Question: How would you know if the segment above had been TRISECTED ?

19


Related Vocabulary


BISECT BISECTOR


(verb) to divide into two congruent parts

(noun) the RAY that divides an angle into two congruent angles

Question: Is it possible for an angle to have a MIDPOINT?Question: How would you know the angle above had been TRISECTED ?

20


Related Vocabulary


TRISECT -

TRISECTORSTRISECTION POINTS


(verb) to divide a segment or angle into THREE congruent parts.

21

1.6 Write paragraph proofs

Related Vocabulary


COUNTEREXAMPLE -

PARAGRAPH PROOF -

Chapter 1, Section 6: “Paragraph Proofs”

Facts that are inconsistent with theory – or an argument proving that a fact, hypothesis or mathematical theorem is not true.

NOTE: This is an introduction to the paragraph method of proof. We will use the Paragraph form

exclusively when we get to Indirect Proofs in Chapter 5.While paragraph proofs can also be used to prove a

mathematical conclusion, you will mostly rely upon the two-column method to do so in this course.

When writing an “Indirect Proof” in paragraph form, you will be attempting to arrive at a conclusion

by proving the alternative to it false.

Therefore, “Indirect Proof” can also be referred to as “Proof by Contradiction.”

Has THREE parts:* Introduction

* Body

* Conclusion

Like any good paper,

22

1.7 Recognize that geometry is based on a deductive structure1.7 Identify undefined terms, postulates, and definitions

1.7 Understand the characteristics and application of theorems

Related Vocabulary


CONCLUSION

CONDITIONAL STATEMENT

CONVERSE

DEDUCTIVE STRUCTUREDEFINITION

HYPOTHESIS

IMPLICATION

POSTULATE

Chapter 1, Section 7: “Deductive Structure”

23



Related Vocabulary


DEDUCTIVE STRUCTURE

DEFINITION

POSTULATE


– conclusions are supported and proved by using allowable assumptions and statements that have been proved to be true.

Deductive reasoning – the process of drawing a conclusion based on logical or reasonable

information or facts.

Inductive reasoning – reaching a conclusion based on observation

alone. Generalizing.

– an unproved assumption. – states the meaning of a term or idea.

UNDEFINED TERMS

– terms that are described. Example: points and lines

24

Use these + theorems in

proofs!



Related Vocabulary


CONCLUSION -

CONDITIONAL STATEMENT -

HYPOTHESIS -

IMPLICATION


DECLARATIVE STATEMENT -

(definition) – a midpoint is a point that divides a segment (or an arc) into two congruent parts

If a point is the midpoint of a segment,

then the point divides the segment into two congruent segments


“If a point is the midpoint of a segment,

The “If . . .,” clause of the conditional statement

The “then . . .” clause of the conditional statement

then the point divides the segment into two congruent segments.”

“If . . ., then . . .”

25



Related Vocabulary


CONCLUSION -


CONVERSE -

HYPOTHESIS -

IMPLICATION


If p, then qIf p,

then q

If q, then p

Let p = “If a point is the midpoint of a segment,”

Let q = “then the point divides the segment into two congruent segments”

If a point divides a segment into two congruent segments, then the point is the midpoint of the segment

< - - - - - - - - - - - - - - - - - >

Reversing the hypothesis and

conclusion

In this definition, the hypothesis and conclusion are

reversible. If a definition is a GOOD

definition, it is always reversible!

26



Related Vocabulary


CONCLUSION -


CONVERSE -

HYPOTHESIS -

IMPLICATION


Theorem 1: If two angles are right angles, then they are congruentIf p,

then q

If q, then p

Let p = “If two angles are right angles,”

Let q = “then they are congruent”

If two angles are congruent, then they are right angles.

< - - - - - - - - - - - - - - - - - >

The converse is FALSE! Postulates and theorems

are NOT always reversible, unlike GOOD

definitions!

Reversing the

hypothesis and

conclusion 27

If you write a definition and find it is false when

reversed, then what you wrote is NOT a GOOD definition!

28

1.8 Recognize conditional statements

1.8 Use the chain-rule to draw conclusions

Related Vocabulary


CONTRAPOSITIVEINVERSE

CHAIN RULE

NEGATION

VENN DIAGRAM

Chapter 1, Section 8: “Statements of Logic”

Also, from 1.7

• Declarative sentence• Conditional sentence• Hypothesis• Conclusion• Implication

1.8 Recognize the negation of a statement1.8 Recognize the converse, the inverse, and the contrapositive of a statement

29



Related Vocabulary


NEGATION -


• Declarative sentence

• Conditional sentence

• Hypothesis

• Conclusion

• Implication


Two straight angles are congruent

If two angles are straight angles, then they are

congruentIf two angles are straight angles,

then they are congruent

If p, then q Symbols: p q

Words: p implies q

Words: “not

p”

Symbols: ~ p

To contradict or state the opposite of something 30



Related Vocabulary


CONTRAPOSITIVE

INVERSE

CONVERSE

VENN DIAGRAM



Conditional “if p, then q”: If you live in Lexington, then you live in Kentucky.

If q, then p:

If you live in Kentucky, then you live in Lexington.

If ~p, then ~q

If you don’t live in Lexington, then you don’t live in Kentucky.

If ~q, then ~p

If you don’t live in Kentucky, then you don’t live in Lexington.

KentuckyLexington

To determine the truth value of each statement, we must first assume that the original conditional statement is

TRUE.

FALSE!FALSE !

TRUE!

If the conditional statement is TRUE,

then the contrapositive will always be TRUE!

31



Related Vocabulary


CHAIN RULE



- The logical sequence you follow when writing proofs

Words: If p implies q, and q implies r, then p implies r.

Symbols: If p q, and q r, then p r.

Mathematically:

since 5 = 5, . . . then x = y

In a Proof:

If ∡X is a right angle and ∡Y is a right angle,

then ∡ X ≅ ∡ Y

and all right angles equal 90,

32

1.9 Solve probability problems

Related Vocabulary


PROBABILITY -

Chapter 1, Section 9: “Probability”

The chance that something will happen

Favorable PART

TOTAL Possibilities

A ratio whose value is

between 0 and 1,

inclusive.

:

Impossible

Certain

0 1½ Equally Likely

Less likely

More Likely

STEPS:

1. List ALL outcomes2. Record

“winners” over total

33

C

B

A

D

If three of the four points are selected in a random order, what is the probability that the ordered letters will correctly name the angle shown?

LIST all possibilities:ABC

ABD

ACB

ACD

ADB

ADC

BAC

BAD

BCA

BCD

BDA

BDC

CAB

CAD

CBA

CBD

CDA

CDB

DAB

DAC

DBA

DBC

DCA

DCB

Or use the Fundamental Counting Principle:

4 3 2# of ways to select the first

point

# of ways to select

the second point

# of ways to select the third

point

TOTAL

24

34

C

B

A

D

If three of the four points are selected in a random order, what is the probability that the ordered letters will correctly name the angle shown?

Circle the “winners”:ABC

ABD

ACB

ACD

ADB

ADC

BAC

BAD

BCA

BCD

BDA

BDC

CAB

CAD

CBA

CBD

CDA

CDB

DAB

DAC

DBA

DBC

DCA

DCB

PART

4

Answer:

Part

TOTAL

24

4

6

1

Don’t forget

to REDUCE

!

35

page 53, chapter summary: concepts and procedures after studying this chapter, you should be able...

Documents

measure segments

measurement of segments

trisectors of segments

noncollinear points1

chapter summary

paragraph proofs1

column proofs1

space bar