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Geometry Notes - Chapter 2
Reasoning and Proof
Warm -up
1. Evaluate each expression for the given value of n.
n( n  3)
a) 3n – 2; n = 4
b)
;n=8
2
c) n2 – 3n; n = 3
2. Solve each equation
a) 6x – 42 = 4x
b) 3x + 4 =
1
x–5
2
c) 2 – 2x =
3. If mAGB = 4x + 7 and mEGD = 71, find x..
A
2
x–2
3
B
F
C
G
4. If mBGC = 45, mCGD = 8x + 4, and
mDGE = 15x – 7, find x..
E
D
2.1 Inductive Reasoning and Conjecture
Vocabulary
An educated guess based on known information is called a __________________.
Examining several specific situations or examples to arrive at a conjecture is called
_________________________.
Think of a career that uses inductive reasoning and state why._______________________
_________________________________________________________________________
Examples:
1. Describe the pattern and make a conjecture about the next number in the pattern:
2, 4, 12, 48, 240
2. Describe the pattern and make a conjecture about the next number in the pattern:
-5, 10, -20, 40
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3. Make a conjecture based on the given information. Draw a figure to illustrate your conjecture.
a) 1 and 2 form a right angle.
b) ABC and DBE are vertical angles.
c) E and F are right angles.
What is a counterexample? ________________________________________________
4. Determine whether each conjecture is true or false. Give a counterexample for any false
conjecture.
a) Given: S, T, and U are collinear and ST = TU.
Conjecture: T is the midpoint of SU.
b) Given: 1 and 2 are adjacent angles.
Conjecture: 1 and 2 form a linear pair.
2.1 Review
1. Make a conjecture based on the given information. Draw a figure to illustrate your conjecture.
a) Point P is the midpoint of NQ .
b) Points A, B, and C are collinear, and D is between B and C.
.
2. Determine whether the conjecture is true or false. Give a counter example for any false
conjecture.
a) Given: ABC and CBD form a linear pair.
Conjecture: :ABC  CBD
b) Given: AB + BC = AC
Conjecture: AB = BC
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2.2 Logic
Vocabulary
Any sentence that is either true or false, (not both) is called a _____________________.
Write down a statement____________________________________________________.
Whether the statement is true or false is called its ___________________.
The negation of a statement has the _____________meaning as well as an ___________
truth value.
Write the negation of the statement written above. ______________________________
_______________________________________________________________________
If p represents some statement, then _________________ is the negation of the statement.
p: ______________________________________________________________________
~p: _____________________________________________________________________
Two or more statements can be joined to form a ________________ statement.
p: ______________________________________________________________________
q: ______________________________________________________________________
A compound statement formed by joining two or more statements with the word ______ is
called a _____________________. This is shown ___________ and read ______________
p  q: ____________________________________________________________________
Example:
1. Use the following statements to write a compound statement for each conjunction. Then find its
truth value.
p: one foot is 14 inches.
q: September has 30 days.
r: Three points on the same line are collinear.
a) p and q
b) r  p
c) ~q  r
d) ~p  r
The only time a conjunction is true is when ______________________________________.
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A compound statement joined by the word _________ is called a ___________________.
This is shown __________________ and read _______________.
Example:
1. Use the following statements to write a compound statement for each disjunction. Then find its
truth value.
p: 8 + 3 = 10.
q: A cow is a mammal
r: Vertical angles are congruent
a) p or q
b) r  p
c) ~q  r
d) ~p  r
A disjunction is true when ____________________________________________________.
Venn Diagrams
Venn diagrams can be used to represent real world problems involving compound statements.
Example:
1. The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap,
jazz, and ballet classes.
a) How many students are enrolled in all
three classes?
Tap
28
b) How many students are enrolled in tap or
ballet?
c) How many students are enrolled in jazz
and ballet and not tap?
Jazz
43
13
9
17
25
Ballet
29
2. There are 120 students in the Sophomore class. Forty students play in the band, sixty-five
students are out for sports, and twenty-five students participate in band and sports. Draw a Venn
Diagram to illustrate the situation.
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Truth Tables
A convenient method for organizing the truth values of statements is a ______________
1. Construct a truth table for ~p  q
p
q
~p
~p  q
2. Construct a truth table for ~p  ~q
p
q
~p
~q
~p  ~q
When constructing truth tables, create a column for ______________________________
________________________________________________________________________
________________________________________________________________________
3. Construct a truth table for p  (~q  r)
2.2 Review
1.
p: -3 – 2 = -5
q: Vertical angles are congruent
r: 2 + 8  10
Write the following compound statements. Then find its truth value.
a) (q  r)
b) (~p  ~r).
2. Construct a truth table for ~p  q
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2.3 Conditional Statements
Vocabulary
In your own words, define the word conditional ________________________________________
What do you think a conditional statement is? __________________________________________
_______________________________________________________________________________
An ___________________________ is written in the form _______________________________.
Where statement p is the ______________ and statement q is the __________________________.
Examples:
1. Identify the hypothesis and conclusion of each statement.
a) If you buy a car, then you get $1500 cash back.
b) The Tigers don’t have practice Saturday morning if they win the football game Friday night.
2. Identify the hypothesis and conclusion of each statement. Then write each statement in if-then
form.
a) A five-sided polygon is a pentagon.
b) An angle with a measure greater than 90 is an obtuse angle.
3. Determine the truth value of the following statement for each set of conditions.
If Matthew rests for 10 days, his ankle will heal.
a) Matthew rests for 10 days, and his ankle still hurts.
b) Matthew rests for 3 days, and his ankle still hurts.
c) Matthew rests for 10 days, and his ankle doesn’t hurt anymore.
d) Matthew rests for 7 days, and his ankle doesn’t hurt anymore.
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Related conditionals
Statement
Formed by
Symbols
Examples
Conditional
Converse
Inverse
Contrapositive
The truth values for the converse and inverse are___________________________________
The truth values for the conditional and the contrapositive are ________________________
p
q
~p
~q
Conditional
pq
Converse
qp
Inverse
~p  ~q
Contrapositive
~q  ~p
Examples:
1. Write the converse, inverse, and contrapositive of the statement:
Linear pairs of angles are supplementary.
Conditional: ____________________________________________________________
Converse: ______________________________________________________________
Inverse: _______________________________________________________________
Contrapositive: _________________________________________________________
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2. Write the converse, inverse, and contrapositive of the statement:
If I have a dog, then I have a pet.
Conditional: ____________________________________________________________
Converse: ______________________________________________________________
Inverse: _______________________________________________________________
Contrapositive: _________________________________________________________
2.3 Review
1. Identify the hypothesis and conclusion of the statement.
If the drama class raises $2000, then they will go on tour.
2. Write the statement in if-then form.
An acute angle has a measure less than 90.
3. Determine the truth value of the statement for each set of conditions.
If you finish your homework by 5 P.M. then you go out to dinner.
a) You finish your homework by 5 P.M., and you go out to dinner.
b) You finish your homework by 4 P.M. and you go out to dinner.
c) You finish your homework by 5 P.M., and you do not go out to dinner.
d) You finish your homework at 7 P.M., and you go out to dinner.
4. Write the inverse of the conditional statement. Determine whether the statement is true or false.
Provide a counterexample if the statement is false.
If 89 is divisible by 2, then 89 is an even number.
2.4 Deductive Reasoning
Vocabulary
Define inductive reasoning from Section 2.1 ______________________________________
__________________________________________________________________________
The process of using facts, rules, definitions, or properties to reach logical conclusions is
called ____________________________________.
The form of deductive reasoning that is used to draw conclusions from _______ conditional
statements is called _______________________________.
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Law of Detachment
If p  q is true and p is true, then q is also true
Examples:
1. Use the Law of Detachment to determine whether each conclusion is valid based on the true
conditional given. If not, write invalid. Explain your reasoning.
If two angles are complementary to the same angle, then the angles are congruent.
a) Given: A and C are complementary to B.
Conclusion: A is congruent to C.
b) Given: A  C
Conclusion: A and C are complements of B.
c) Given: E and F are complementary to G.
Conclusion: E and F are vertical angles.
Law of Syllogism
If p  q and q  r are true, then p  r is also true
Examples:
2. Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set
of statements. If a valid conclusion is possible, write it. If not, write no conclusion.
a) If two angles form a linear pair, then the two angles are supplementary.
If two angles are supplementary, then the sum of their measures is 180.
b) If a hurricane is Category 5, then winds are greater than 155 miles per hour.
If winds are greater than 155 miles per hour, then trees, shrubs, and signs are blown down.
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3. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment
or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
a) (1) If a whole number is even, then its square is divisible by 4.
(2) The number I am thinking of is an even whole number.
(3) The square of the number I am thinking of is divisible by 4.
b) (1) If the football team wins their game Friday then Jacob will go out with his
friends on Saturday.
(2) Jacob goes out with is friends on Saturday.
(3) The football team won their game Friday.
2.4 Review
1. Determine whether the stated conclusion is valid based on the given information. If not, write
invalid. Explain your reasoning. If two angles have a sum of 180, then they are supplementary.
1. Given: mA + mB is 180.
Conclusion: A and B are supplementary
2. Use the Law of Syllogism to determine whether a valid conclusion can be reached from the set
of statements. Write out any valid conclusions.
If the heat wave continues, then air conditioning will be used more frequently. If air
conditioning is used more frequently, then energy costs will be higher.
3. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment
or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
(1) If it is Tuesday, then Marla tutors chemistry.
(2) If Marla tutors chemistry, then she arrives home at 4 P.M.
(3) If Marla arrives at home at 4 P.M., then it is Tuesday.
2.5 Postulates and Paragraph Proofs
Vocabulary
A statement, accepted as true, that describes a fundamentals relationship between basic
terms of geometry (like points, lines, and planes) is called a ______________________
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Postulates about points, lines, and planes

Through any two points, there is _______________________.

Through any three points not on the same line, there is ____________________.

A line contains at least ________ points.

A plane contains at least three points not on the same ___________.

If two points lie in a plane, then the entire line containing those points lies _____
__________________

If two lines intersect, then their intersection is a ______________.

If two planes intersect, then their intersection is a ______________.
Examples:
1. Determine the number of segments that can be
drawn connecting each pair of points
2. Determine whether each statement is always, sometimes, or never true. Explain.
a) A line contains exactly 1 point.
b) Noncollinear points R, S, and T are contained in exactly one plane.
c) Any two lines l and m intersect.
d) Planes R and S intersect in point T.
3. In the figure, AC and DE are in plane Q and
AC  ED . State the postulate that can be used
to show each statement is true.
F
a) Exactly one plane contains points F, B, and E.
C
B
A
D
E
Q
G
b) BE lies in plane Q.
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Postulates, along with definitions, and algebraic properties of equality are used to ___________
_____________________________________________________________________________.
Once a statement or conjecture is proven to be true, it is called a _________________________.
A ______________ is a logical argument in which each statement you make is ______________
_____________________________________________________________________________.
Two types of proofs are ________________________ and _____________________________.
The 4 Essential parts of a good proof are:
1.
2.
3.
4.
In a _______________________proof you write a paragraph to explain why a conjecture for a given
situation is true.
4. Given that M is the midpoint of PQ , write a paragraph proof to show that PM  MQ .
5. Given AC intersecting CD , write a paragraph proof to show that A, C, and D determine a
plane.
2.5 Review
1. Determine whether the following statements are always, sometimes, or never true. Explain.
a) Three collinear points determine a plane.
b) Two points A and B determine a line.
c) A plane contains at least three lines.
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2. In the figure, lines DG and DP lie in plane J and H lies on line
DG. State the postulate that can be used to show each statement
is true.
a) G and P are collinear.
P
H
D
J
G
b) Points D, H, and P are coplanar.
2.6 Algebraic Proof
Properties of Equality for Real Numbers
Reflexive Property
For every number a, ____________________
Symmetric Property
For all numbers a and b, __________________________________
Transitive Property
For all numbers a, b, and c, ________________________________
Addition and
Subtraction Properties
For all numbers a, b, and c, ________________________________
Multiplication and
Division Properties
For all numbers a, b, and c, ________________________________
Substitution Property
For all numbers a and b, __________________________________
Distributive Property
For all numbers a, b, and c, ________________________________
Properties of equality can be used to ____________________________________________
_________________________________________________________________________.
A group of algebraic steps used to solve problems form a ___________________________.
Examples:
1. Solve the problem and justify each step.
2(5 – 3a) – 4(a + 7) = 92
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2. State the property that justifies each step.
a) If 7d + 3 = 24, then 7d = 21
b) If 7 – 2n + n = 2 – 5n + 8, then 7 – n = 10 – 5n
c) If 3x = 6, then x = 2
3
Complete the two-column proof:
Given:
4x  6
=9
2
Prove: x = 3
Statements
4x  6
a.
=9
2
 4x  6 
b. ____ 
 = 2(9)
 2 
Reasons
a.
c. 4x + 6 = 18
c.
d. 4x + 6 – 6 = 18 – 6
d.
e. 4x = _______
e.
f.
g,
4x
=
4
b.
f.
g.
4. Write a two-column proof:
Given: 4x + 8 = x + 2
Prove: x = -2
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5. Write a two-column proof:
Given: mABC + mCBD = 90,
mABC = 3x – 5, and
x 1
mCBD =
,
2
Prove: x = 27
D
C
B
A
2.6 Review
State the property that justifies each statement.
1. If 80 = mA. then mA = 80.
2. If RS = TU and TU = YP, then RS = YP.
3. If 7x = 28, then x = 4
4. If VR + TY = EN + TY, then VR = EN
5. If m1 = 30 and m1 = m2, then m2 = 30
2.7 Proving Segment Relationships
Segment Congruence Properties and Postulates
Property, Postulate, or Definition
Segments
Reflexive Property
AB  ___________
Symmetric Property
If AB  CD , then _________
Transitive
Property
If AB  CD and CD  EF , then________________
Definition of congruence
If AB = BC, then _______________
Segment Addition Postulate
If B is between A and C, then __________________
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Examples:
1. Justify each statement with a property of equality or a property of congruence.
a) QA = AQ
b) If AB  BC and BC  CE , then AB  CE
c) If Q is between P and R, then PR = PQ + QR
d) If AB + BC = EF + FG and AB + BC = AC, then EF + FG = AC
2. Complete each proof.
Given: SU  LR
TU  LN
Prove: ST  NR
T
S
L
U
N
Statements
a. SU  LR , TU  LN
b.
c. SU = ST + TU
LR = LN + NR
d. ST + TU = LN + NR
e. ST + LN = LN + NR
f. ST + LN – LN = LN + NR – LN
g.
R
Reasons
a.
b. Definition of  segments
c.
d.
e.
f.
g. Substitution Property
3. Complete each proof.
Given: AB  CD
Prove: CD  AB
Statements
a.
b. AB = CD
c. CD = AB
d.
4. Write a two-column proof.
Given: Q is the midpoint of PR
R is the midpoint of QS
Prove: PR = QS
Reasons
a. Given
b.
c.
d. Definition of  segments
S
R
Q
P
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2-7 Review
Justify each statement with a property of equality, a property of congruence, or a postulate.
1. If XY = YZ, then XY  YZ
2. If QR = ST and ST = 5, then QR = 5
 
3. If QR = ST and 5 = ST, then QR = 5
4. If AB = CD, then AB + BC = CD + BC
2.8 Proving Angle Relationships
Property, Postulate, or
Definition
Reflexive Property
Angle Congruence Properties, Postulates, and Theorems
Angles
1  ___________
Symmetric Property
If 1  2, then ___________
Transitive Property
If 1  2 and 2  3, then___________
Definition of congruence
1  2, then _______________
Protractor Postulate
Given AB and a number r between 0 and 180, there is exactly one
ray with endpoint A, extending on either side of AB , such that the
measure of the angle formed is r.
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Example
Theorem or Postulate
Angle Addition Postulate
If R is in the interior of PQS, then
_________________________________
Supplement Theorem
If two angles form a linear pair, then they
are supplementary.
Complement Theorem
If the noncommon sides of two adjacent
angles form a right angle, then _______
________________________________
Angles supplementary to the same angle
or to congruent angles _____________.
Angles complementary to the same angle
or to congruent angles _____________.
Vertical Angles Theorem
If two angles are vertical angles, then
______________________________
Right Angle Theorems
Perpendicular lines intersect to form ___________________________________
All right angles are ___________________
Perpendicular lines form _____________________________________________
If two angles are congruent and supplementary, then _________________________
If two congruent angles form a linear pair, then _____________________________
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Examples:
1. Find the measure of each numbered angle.
a) m2 = 57
b) m3 = 38
4
1
3
2
6
c) 9 and 10 are complementary.
7  9, m8 = 41
5
8 9
7
10
2. Determine whether the following statements are always, sometimes, or never true.
a) Two angles that are supplementary form a linear pair.
b) Two angles that are vertical are adjacent.
3. Write a two-column proof
Given: 1 and 2 form a linear pair.
2 and 3 are supplementary.
Prove: 1  3
1
Statements
Reasons
2
3
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