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Chapter 2:
2-1:
Deductive Reasoning
If - Then Statements
IF - THEN STATEMENT: a statement whose basic form is If
also known as
example:
(page 32)
(page 33)
, then
.
statements or
.
If it is snowing, then it is cold.
Conditional statements are either
or
HYPOTHESIS (p): a statement that follows the “
.
” in an “If - then” statement.
example:
CONCLUSION (q): a statement that follows the “
” in an “If - then” statement.
example:
p⇒q
Forms for Conditional Statements:
If it is snowing, then it is cold.
If p, then q.
It is snowing implies it is cold.
p implies q.
It is snowing only if it is cold.
p only if q.
It is cold if it is snowing.
q if p.
Circle one: TRUE or FALSE
NOTE: If, then, implies, & only if are
part of the hypothesis or the conclusion.
example: If angle X is acute, then the measure of angle X equals 60 degrees.
____________________________________ implies ___________________________________
____________________________________ only if ____________________________________
______________________________________ if _______________________________________
Circle one: TRUE or FALSE
CONVERSE (of a conditional statement): is formed by interchanging the
and the conclusion.
Converses to conditional statements are either
or
.
example:
Conditional: If it is snowing, then it is cold.
Converse:
If p , then q .
, then
If
. If
, then
.
Circle one: TRUE or FALSE
example:
Conditional: If angle X is acute, then the measure of angle X equals 60 degrees.
Converse:
, then
If
.
Circle one: TRUE or FALSE
You can not assume a converse is
just because the original statement is true.
COUNTEREXAMPLE: an example used to prove that an if-then statement is
The hypothesis is
NOTE: only
example:
and the conclusion is
.
.
counterexample is needed to prove a statement false!
Two angles are adjacent if they have a common vertex.
Conditional: If
, then
.
TRUE or FALSE Converse: If
, then
.
TRUE or FALSE -
BICONDITIONAL: a statement that combines a
with the words “if and only if”.
and its
example:
Conditional:
If two angles are congruent, then their measures are equal.
p⇒q
Biconditional:
if and only if
p⇔ q
Assignment: Written Exercises, page 35: 3,5,7,9,12,15,16,18,21,24,27,30,31
.
2-2:
Properties from Algebra
Properties of Equality (=)
Addition Property:
Subtraction Property:
Multiplication Property:
Division Property:
Substitution Property:
Reflexive Property:
Symmetric Property:
Transitive Property:
Distributive Property:
Properties of Congruence ( ! )
Reflexive Property:
Symmetric Property:
Transitive Property:
(page 37)
Statement
Justification
a. If AB = CD and BC = BC, then AB + BC = CD + BC.
b. If 2 + YZ = 8, then YZ = 6.
c. If 2 m∠1 = 72º, then m∠1 = 36º.
d. If Pt.B is between A and C, then AB + BC = AC.
e. If m∠A = m∠X and m∠X = m∠B, then m∠A = m∠B.
f. If AB
!
g. If ∠A
! ∠B, then ∠B ! ∠A.
CD and CD
!
EF , then AB
!
EF .
h. If m∠K = 90º, then ∠K is a right angle.
Solve for “x” and supply reasons for each step:
Steps
1.
5(3x-4)=5x
Reasons
1.
2.
2.
3.
3.
4.
4.
5.
5.
Complete each of the following proofs.
Given: m∠1 = m∠3
D
E
Prove: m∠ABE = m∠DBC
1
A
Statements
1.
2.
2.
3.
3.
+
m∠DBC =
+
B
3
C
Reasons
1.
4. m∠ABE =
2
Given
4.
5.
5.
Given: KP = ST ; PR = TV
K
P
R
S
T
Prove: KR = SV
Statements
Reasons
1.
1.
2.
2.
3. KP + PR =
3.
Given
ST + TV =
4.
4.
Assignment: Written Exercises, pages 41 & 42: 1 to 13 odd numbers
V
Proving Theorems
2-3:
(page 43)
What is a theorem?
THEOREM 2-1
MIDPOINT THEOREM
If M is the midpoint of AB , then AM =
AB and MB =
AB.
Given: M is the midpoint of AB
Prove:
A
M
B
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
example: Given R is the midpoint of SQ. Give the reason that justifies each statement.
!
RQ
(a)
SR
(b)
SR = 1/2 SQ
S
R
(c)
SR + RQ = SQ
(d)
PR bisects SQ
_
P
Q
THEOREM 2-2
ANGLE BISECTOR THEOREM
!!!"
If BX is the bisector of ∠ABC, then: m∠ABX =
and
m∠ABC
m∠XBC =
m∠ABC.
!!!"
Given: BX is the bisector of ∠ABC.
A
Prove:
X
B
C
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
example:
!!!"
Given: FD bisects ∠CFE. Give the reason that justifies each statement.
C
(a)
m∠CFD = ½ m∠CFE
(b)
m∠CFD = m∠DFE
(c)
CD + DE = CE
D
F
Reasons Used in Proofs:
(1)
(2)
(3)
(4)
Given information
Definitions
Postulates (including properties from algebra)
Theorems - only ones that have already been proved!
E
DEDUCTIVE REASONING: proving statements by reasoning from accepted postulates,
,
, and given information.
example:
NOTE: Definitions can be written as biconditionals (combine conditional and converse),ie. the
conditional and converse are both
.
Conditional:
Converse:
If an angle is a right angle,then its measure is 90º.
If the measure of an angle is 90º, then the angle is a right angle.
Biconditional:
An angle is a right angle
its measure is 90º.
example:
Given: M is the midpoint of PQ
N is the midpoint of RS
PQ = RS
P
M
Q
Prove: PM = RN
R
N
S
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Assignment: Written Exercises, pages 46 & 47: 1-14 all, 19 & 20
Worksheet on Lesson 2-3
Prepare for Quiz on Lessons 2-1 to 2-3: Using Deductive Reasoning
2-4:
Special Pairs of Angles
(page 50)
COMPLEMENTARY ANGLES: two angles whose measures have the sum
degrees.
example:
SUPPLEMENTARY ANGLES: two angles whose measures have the sum
degrees.
example:
examples:
B
A
(1)
X
C
In the diagram, ∠CXD is a right angle.
E
Name another right angle:
D
Name complementary angles:
Name 2 congruent supplementary angles:
Name 2 noncongruent supplementary angles:
(2)
∠C and ∠D are complementary, m∠C = 3 x - 5 and m∠D = x + 15.
Find the value of x, m∠C, and m∠D.
x =
m∠C =
m∠D =
(3)
∠P and ∠K are suppplementary, m∠P = 4 x + 10 and m∠K = x + 20.
Find the value of x, m∠P, and m∠K.
x =
m∠P =
m∠K =
(4)
A supplement of an angle is seven times a complement of the angle.
Find the measures of the angle, its complement, and its supplement.
measure of angle =
=
measure of comp. =
=
measure of supp. =
=
(5)
A complement of an angle is 15 less than twice the measure of the angle.
Find the measures of the angle, its complement, and its supplement.
measure of angle =
=
measure of comp. =
=
measure of supp. =
=
VERTICAL ANGLES: two angles whose sides form two pairs of
rays.
example:
1
4
2
3
THEOREM 2-3
Vertical angles are congruent.
Given: ∠1 and ∠2 are vertical angles.
3
1
Prove:
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
2
examples: Find the value of “x”.
(4)
(5)
35º
35º
(2x-5)º
xº
x=
x=
(6)
(7)
(x+30)º
(2x+30)º
(x+y)º
(4x)º
(4y)º
x=
120º
x=
y=
(8)
Find the measure of each angle.
C
D
60º
A
B
G
Assignment: Written Exercises, pages 52 to 54: 1-31 odd #’s, 32-35
Worksheet on Lesson 2-4
Prepare for Quiz on Lesson 2-4: Special Pairs of Angles
E
F
Perpendicular Lines
2-5:
(page 56)
PERPENDICULAR LINES (⊥−lines): two lines that intersect to form
angles.
example:
1 2
4 3
m
n
If m ⊥ n , then
,
,
, and
are right angles.
How many angles must be right angles in order for the lines to be perpendicular?
THEOREM 2-4
If two lines are perpendicular, then they form
adjacent angles.
l
Given: l ⊥ n
Prove:
,
,
are congruent angles.
2 1
3 4
,&
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
n
THEOREM 2-5
If two lines form congruent adjacent angles, then they are
.
l
Given: ∠1 ≅ ∠2
Prove: l ⊥ n
2 1
n
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
THEOREM 2-6
If the exterior sides of two adjacent acute angles are
, then the angles are complementary.
Given:
!!!" !!!"
OA ! OC
A
Prove: _________________________
B
Proof:
O
Statements
C
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
example:
Name the definition or theorem that
justifies each statement about the diagram.
B
1
2
3
4
5
6
7
8
A
C
D
(a) If AB ! BC , then ∠ABC is a right angle.
(b) If BD ! AC , then ∠3 ≅ ∠4.
(c) If CD ! AD , then ∠7 and ∠8 are complementary.
(d) If ∠7 & ∠8 are complementary, then m∠7 + m∠8 = 90º.
(e) If ∠4 ≅ ∠6, then AC ! BD .
(f) ∠ 4 ≅ ∠5.
(g) If ∠ABC is a right angle, then m∠ABC = 90º.
!!!" !!!"
example: If ZW ! ZY , m∠1 = 5 x, and m∠2 = 2 x - 1, find the value of “x”.
x = ________________
W
V
1
2
Z
Y
Assignment: Written Exercises, pages 58 & 59: 3-11 odd #’s, 15-25 odd #’s
Prepare for Quiz on Lessons 2-2 to 2-5
2-6:
Planning a Proof
(page 60)
A proof of a theorem consists of five (5) parts:
1.
of the theorem.
2. A
that illustrates the given information.
3. A list, in terms of the figure, of what is
.
4. A list, in terms of the figure, of what you are to
5. A series of
.
and
the
that lead from
information to the statement that is to be
.
Methods for Writing Proofs:
A.
as much information as you can.
Sometimes what you can see will show you a plan.
Reread the given. What does it tell you?
Look at the diagram. What other information can you conclude?
B.
Work
Think:
. Go to the conclusion, the part you would like to prove.
“This conclusion would be true if
“And
? would be true if
?
Suggestions for Writing Proofs:
? ”.
”. And so on, until you have a plan.
(1)
Follow one of the suggested methods.
(2)
(3)
(4)
Study
Use a pencil.
TRY, TRY, TRY!!!
proofs.
NOTES
There is usually more than one way to write a proof of a statement.
Be sure that your steps flow logically from one step to the next.
Some steps are more important than others.
Your proof should have enough steps to allow the reader to follow your argument
and to see that the statement is indeed true.
THEOREM 2-7
If two angles are
of congruent angles
(or of the same angle), then the two angles are congruent.
Given: ∠1 and ∠2 are supplementary;
∠3 and ∠4 are supplementary;
∠2 ≅ ∠4
1
2
Prove:
3
4
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
example:
Given: ∠2 is supplementary to ∠3
Prove: ∠1 ≅ ∠3
1
2
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
3 4
THEOREM 2-8
If two angles are
of congruent angles
(or of the same angle), then the two angles are congruent.
Given: ∠1 and ∠2 are complementary;
∠3 and ∠4 are complementary;
∠2 ≅ ∠4
1
2
3
4
Prove:
Proof:
Statements
Reasons
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
example:
Given: LM ! MN and KN ! MN
M
N
1
a. Name a complement of ∠2:
2
3
4
b. Name a complement of ∠3:
L
c. What theorem justifies the answers to a & b?
d. If ∠2 ≅ ∠3, what can be concluded and why?
Assignment: Written Exercises, pages 63 & 64: 1-21 odd #’s
Prepare for Quiz on Lessons 2-4 to 2-6
Prepare for Test on Chapter 2: Deductive Reasoning
K