Page 53, Chapter Summary: Concepts and Procedures
After studying this CHAPTER, you should be able to . . .
1.1
1.2
1.2
1.2
1.3
1.3
1.3
1.3
1.4
1.5
1.6
1.7
1.7
1.7
1.8
Recognize points, lines, segments, rays, angles, and triangles.
Measure segments and angles
Classify angles and name the parts of a degree
Recognize congruent angles and segments
Recognize collinear and noncollinear points
Recognize when a point is between two other points
Apply the triangle inequality principle
Correctly interpret geometric diagrams
Write simple two-column proofs
Identify bisectors and trisectors of segments and angles
Write paragraph proofs
Recognize that geometry is based on a deductive structure
Identify undefined terms, postulates, and definitions
Understand the characteristics and application of theorems
Recognize conditional statements and the negation, the converse,
the inverse, and the contrapositive of a statement
1.8 Use the chain-rule to draw conclusions
1.9 Solve probability problems
2
Chapter 1, Section 1: “Getting Started”
After studying this SECTION, you should be able to . . .
1.1 Recognize points, lines, segments, rays, angles, and triangles.
Related Vocabulary
POINT
VERTEX
INTERSECTION
LINE
ANGLE
TRIANGLE
UNION
SEGMENT
LINE SEGMENT
ENDPOINTS
NUMBER LINE
RAY
3
Chapter 1, Section 1: “Getting Started”
After studying this SECTION, you should be able to . . .
1.1 Recognize points, lines, segments, rays, angles, and triangles.
Related Vocabulary
POINT
LINE
ANGLE
SEGMENT
LINE SEGMENT
ENDPOINTS
4
Chapter 1, Section 1: “Getting Started”
After studying this SECTION, you should be able to . . .
1.1 Recognize points, lines, segments, rays, angles, and triangles.
Related Vocabulary
VERTEX
INTERSECTION
RAY
UNION
TRIANGLE
NUMBER LINE
-3
-2
-1
0
1
2
3
5
Your Turn!
What’s My Name?
To see answers, hit space bar.
Q
1.
2.
C
3.
A
D
point Q
or
Q
ray CA
or
ray CT
T
CA
line DG
G
O
DG
4.
B
E
CT
…or line DO, GD, GO, or OD
DO
segment BE
BE
GD
GO
OD
or segment EB
EB
Easy peasy!
6
Chapter 1, Section 2: “Measurement of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.2 Measure segments and angles
1.2 Classify angles and name the parts of a degree
1.2 Recognize congruent angles and segments
Related Vocabulary
ACUTE ANGLE
OBTUSE ANGLE
RIGHT ANGLE
STRAIGHT ANGLE
CONGRUENT ANGLES
CONGRUENT SEGMENTS
MEASURE
DEGREES
MINUTES
SECONDS
PROTRACTOR
TICK MARK
7
Chapter 1, Section 2: “Measurement of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.2 Measure segments and angles
1.2 Classify angles and name the parts of a degree
1.2 Recognize congruent angles and segments
Related Vocabulary
ACUTE ANGLE
OBTUSE ANGLE
RIGHT ANGLE
STRAIGHT ANGLE
8
Chapter 1, Section 2: “Measurement of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.2 Measure segments and angles
1.2 Classify angles and name the parts of a degree
1.2 Recognize congruent angles and segments
Related Vocabulary
CONGRUENT ANGLES
CONGRUENT SEGMENTS
9
Chapter 1, Section 2: “Measurement of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.2 Measure segments and angles
1.2 Classify angles and name the parts of a degree
1.2 Recognize congruent angles and segments
Related Vocabulary
PROTRACTOR
RULER
MEASURE
TICK MARK
360⁰
359⁰ 60’
DEGREES
Degrees &
MINUTES
359⁰ 59’ 60”
Degrees,
Minutes, &
SECONDS
10
Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”
After studying this SECTION, you should be able to . . .
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
Related Vocabulary
X, Y, and Z are NOT collinear
COLLINEAR
X
Y
NONCOLLINEAR
Z
Z
X
X, Y, and Z are collinear
Y
11
Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”
After studying this SECTION, you should be able to . . .
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
Related Vocabulary
COLLINEARITY  Betweenness of Points
Y is NOT between X and Z
X
Y
Z
Z
X
Y is between X and Z
Y
12
Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”
After studying this SECTION, you should be able to . . .
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
POSTULATE:
The sum of the measures of any two sides of a triangle
is always greater than the measure of the third side.
Nope!
YES!
13
Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”
After studying this SECTION, you should be able to . . .
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
TRIANGLE
INEQUALITY:
For any three points, there are only two possibilities:
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!
YES!
The sum of any two sides exceeds the measure of the third side!
14
Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”
After studying this SECTION, you should be able to . . .
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
See very important TABLE on page 19!
Do Assume:
• Straight lines
• Straight angles
• Noncollinearity
• Betweenness of points
• Relative position of points
AD and BE are straight lines
∡BCE is a straight angle
Allowable Assumptions:
B
C, D, and E are noncollinear
C is between B and E
E is to the right of A
C
A
D
E
15
Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions”
After studying this SECTION, you should be able to . . .
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
See very important TABLE on page 19!
DO NOT Assume:
• Right Angles
• Congruent segments
• Congruent Angles
• Relative SIZES of segments
• Relative SIZES of angles
∡BAC is a right angle
AB ≅ CD
∡B ≅ ∡E
Forbidden Assumptions:
B
You must PROVE these!
∡CDE is an obtuse angle
BC is longer than CE
C
A
D
E
16
Chapter 1, Section 4: “Beginning Proofs”
After studying this SECTION, you should be able to . . .
1.4 Write simple two-column proofs
Related Vocabulary
THEOREM - a mathematical statement that can be proved
Example, Theorem 1: If two angles are right angles, then they are congruent.
TWO-COLUMN PROOF - 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
#1 Given: ∡A is a right ∡
∡B is a right ∡
#2 Prove ∡A ≅ ∡B
#4 Statements
A
#3 Diagram
B
17
#5 Reasons
1. ∡A is a right ∡
1. Given
2. m∡A = 90
2. If an ∡ is a right ∡, then
its measure is 90
3. ∡B is a right ∡
3. Given
4. m∡B = 90
4. Same as #2
5. ∡A ≅ ∡B
5. If 2 ∡’s have the same
measure, then they are ≅
Chapter 1, Section 5: “Division of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.5 Identify bisectors and trisectors of segments and angles
Related Vocabulary
BISECT
BISECTOR
MIDPOINT
TRISECT
TRISECTORS
TRISECTION POINTS
18
Chapter 1, Section 5: “Division of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.5 Identify bisectors and trisectors of segments and angles
Related Vocabulary
BISECT
(verb) to divide into two
congruent parts
BISECTOR
(noun) the POINT that
divides a segment into two
congruent segments
MIDPOINT
(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
Chapter 1, Section 5: “Division of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.5 Identify bisectors and trisectors of segments and angles
Related Vocabulary
BISECT
(verb) to divide into two
congruent parts
BISECTOR
(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
Chapter 1, Section 5: “Division of Segments and Angles”
After studying this SECTION, you should be able to . . .
1.5 Identify bisectors and trisectors of segments and angles
Related Vocabulary
TRISECT -
(verb) to divide a segment or angle into THREE congruent parts.
TRISECTORS
TRISECTION POINTS
21
Chapter 1, Section 6: “Paragraph Proofs”
After studying this SECTION, you should be able to . . .
1.6 Write paragraph proofs
Related Vocabulary
COUNTEREXAMPLE -
PARAGRAPH PROOF Like any good paper,
Has THREE parts:
* Introduction
* Body
* Conclusion
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 twocolumn 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.”
22
Chapter 1, Section 7: “Deductive Structure”
After studying this SECTION, you should be able to . . .
1.7 Recognize that geometry is based on a deductive structure
1.7 Identify undefined terms, postulates, and definitions
1.7 Understand the characteristics and application of theorems
Related Vocabulary
CONVERSE
CONDITIONAL STATEMENT
DEDUCTIVE STRUCTURE
IMPLICATION
DEFINITION
HYPOTHESIS
CONCLUSION
POSTULATE
23
Chapter 1, Section 7: “Deductive Structure”
After studying this SECTION, you should be able to . . .
1.7 Recognize that geometry is based on a deductive structure
1.7 Identify undefined terms, postulates, and definitions
1.7 Understand the characteristics and application of theorems
Related Vocabulary
Deductive reasoning – the process of
drawing a conclusion based on logical or
reasonable information or facts.
DEDUCTIVE STRUCTURE
UNDEFINED TERMS
POSTULATE
DEFINITION
Inductive reasoning – reaching a
conclusion based on observation alone.
Generalizing.
– conclusions are supported and proved by using allowable
assumptions and statements that have been proved to be true.
– terms that are described.
Example: points and lines
– an unproved assumption.
Use these +
theorems in proofs!
– states the meaning of a term or idea.
24
Chapter 1, Section 7: “Deductive Structure”
After studying this SECTION, you should be able to . . .
1.7 Recognize that geometry is based on a deductive structure
1.7 Identify undefined terms, postulates, and definitions
1.7 Understand the characteristics and application of theorems
Related Vocabulary
DECLARATIVE STATEMENT - (definition) – a midpoint is a point that divides a segment
(or an arc) into two congruent parts
CONDITIONAL STATEMENT -
If a point is the midpoint of a segment,
then the point divides the segment into two congruent segments
IMPLICATION

CONDITIONAL STATEMENT

“If . . ., then . . .”
HYPOTHESIS - The “If . . .,” clause of the conditional statement 
“If a point is the midpoint of a segment,
CONCLUSION - The “then . . .” clause of the conditional statement 
25
then the point divides the segment into two congruent segments.”
Chapter 1, Section 7: “Deductive Structure”
After studying this SECTION, you should be able to . . .
1.7 Recognize that geometry is based on a deductive structure
1.7 Identify undefined terms, postulates, and definitions
1.7 Understand the characteristics and application of theorems
Related Vocabulary
In this definition, the hypothesis
< - - - - - - - - -are
- - - -reversible.
- - - - > IMPLICATION
CONDITIONAL STATEMENT
and conclusion
If p, thenisqa GOOD
If a definition
HYPOTHESIS - If p,
is midpoint of a segment,”
Let definition,
p = “If a point it
is the
always reversible!
CONCLUSION - then q
Let q = “then the point divides the segment into two congruent segments”
CONVERSE Reversing the
hypothesis and
conclusion
If q, then p
If a point divides a segment into two congruent segments,
then the point is the midpoint of the segment
26
Chapter 1, Section 7: “Deductive Structure”
After studying this SECTION, you should be able to . . .
1.7 Recognize that geometry is based on a deductive structure
1.7 Identify undefined terms, postulates, and definitions
1.7 Understand the characteristics and application of theorems
Related Vocabulary
The converse is FALSE!
Postulates and theorems are
If two angles are right angles, then they are congruent
NOT always reversible, unlike
If p,
Let p = “If two angles are right angles,”
GOOD definitions!
CONDITIONAL STATEMENT < - - - - - - - - - - - - - - - - - > IMPLICATION
Theorem 1:
HYPOTHESIS -
CONCLUSION - then q
Let q = “then they are congruent”
CONVERSE -
If q, then p
If two angles are congruent,
then they are right angles.
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
Chapter 1, Section 8: “Statements of Logic”
After studying this SECTION, you should be able to . . .
1.8
1.8
1.8
1.8
Recognize conditional statements
Recognize the negation of a statement
Recognize the converse, the inverse, and the contrapositive of a statement
Use the chain-rule to draw conclusions
Related Vocabulary
CHAIN RULE
CONTRAPOSITIVE
INVERSE
NEGATION
VENN DIAGRAM
Also, from 1.7
• Declarative sentence
• Conditional sentence
• Hypothesis
• Conclusion
• Implication
29
Chapter 1, Section 8: “Statements of Logic”
After studying this SECTION, you should be able to . . .
1.8
1.8
1.8
1.8
Recognize conditional statements
Recognize the negation of a statement
Recognize the converse, the inverse, and the contrapositive of a statement
Use the chain-rule to draw conclusions
Related Vocabulary
• Declarative sentence
Two straight angles are congruent
• Conditional sentence
If two angles are straight angles,
then they are congruent
• Hypothesis
If two angles are straight angles,
• Conclusion
then they are congruent
• Implication
If p, then q
NEGATION - To contradict or state the
opposite of something
Symbols:
pq
Symbols:
~p
Words:
p implies q
Words:
“not p”
30
Chapter 1, Section 8: “Statements of Logic”
After studying this SECTION, you should be able to . . .
1.8
1.8
1.8
1.8
Recognize conditional statements
Recognize the negation of a statement
Recognize the converse, the inverse, and the contrapositive of a statement
Use the chain-rule to draw conclusions
Related Vocabulary
If the conditional statement is
TRUE,
Conditional “if p, then q”: If you live
in Lexington, then you live in Kentucky.
thenp:the contrapositive will always
If
q,
then
CONVERSE
If you live in Kentucky, then you live in Lexington.
FALSE!
be TRUE!
INVERSE
If ~p, then ~q
F
A
L
If you don’t live in Lexington, then you don’t live in Kentucky. S
E !
CONTRAPOSITIVE If ~q, then ~p
VENN DIAGRAM
To determine the truth value of each
statement, we must first assume that the
original conditional statement is TRUE.
If you don’t live in Kentucky, then you don’t live in Lexington.
Kentucky
TRUE!
Lexington
31
Chapter 1, Section 8: “Statements of Logic”
After studying this SECTION, you should be able to . . .
1.8
1.8
1.8
1.8
Recognize conditional statements
Recognize the negation of a statement
Recognize the converse, the inverse, and the contrapositive of a statement
Use the chain-rule to draw conclusions
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,
and all right angles
equal 90,
then ∡ X ≅ ∡ Y
32
Chapter 1, Section 9: “Probability”
After studying this SECTION, you should be able to . . .
1.9 Solve probability problems
Related Vocabulary
PROBABILITY -
The chance that something will happen
STEPS:
A ratio whose
value is between
0 and 1, inclusive.
:
Favorable PART
TOTAL Possibilities
0
Impossible
1. List ALL outcomes
2. Record “winners”
over total
½
Less
likely
Equally
Likely
1
More
Likely
Certain
33
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?
TOTAL
A
B
C
24
D
LIST all possibilities:
BAC
CAB
ABC
DAB
ABD
BAD
CAD
DAC
ACB
BCA
CBA
DBA
ACD
BCD
CBD
DBC
ADB
BDA
CDA
DCA
ADC
BDC
CDB
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
34
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”:
BAC
CAB
ABC
PART
A
B
C
Don’t
forget to
DAB
REDUCE!
ABD
BAD
CAD
DAC
ACB
BCA
CBA
DBA
ACD
BCD
CBD
DBC
ADB
BDA
CDA
DCA
ADC
BDC
CDB
DCB
4
D
Answer:
Part
41
TOTAL
24
6
35