Transcript Ch 2 Reasoning and Proof

Lesson 2.1
Conditional
Statements
You will learn to…
* recognize and analyze a conditional
statement
* write postulates about points, lines, an
planes using conditional statements
A conditional statement has
two parts, a hypothesis and a
conclusion.
pq
If p, then q.
hypothesis (p)
If the team wins the game,
then they will win the
tournament.
conclusion (q)
Write an if-then statement.
1. The intersection of two
planes is a line.
If two planes intersect, then
their intersection is a line.
Write an if-then statement.
2. A line containing two given
points lies in a plane if the
two points lie in the plane.
If two points lie in a plane,
then the line containing them
lies in the plane.
The converse is formed by
switching the hypothesis and
conclusion.
The converse is q  p.
If q, then p.
Write the converse of this if-then
statement. Is it true or false?
3. If m  A = 125°,
then  A is obtuse.
If  A is obtuse,
then m  A = 125°.
False
The negation of a statement is
formed by negating the
statement.
The negation is written ~ p.
Write the negation of this
statement.
4. m  A = 125°
m  A  125°
5.  A is not obtuse
 A is obtuse
The inverse is formed by
negating the hypothesis and
the conclusion.
The inverse is ~ p  ~ q.
If ~ p, then ~ q.
Write the inverse of this if-then
statement. Is it true or false?
6. If m  A = 125°,
then  A is obtuse.
If m  A  125°,
then  A is not obtuse.
False
The contrapositive is formed
by negating the hypothesis and
conclusion of the converse.
The contrapositive is ~ q  ~ p.
If ~ q, then ~ p.
Write the contrapositive of this ifthen statement. Is it true or false?
7. If m  A = 125°,
then  A is obtuse.
If  A is not obtuse,
then m  A  125°.
True
Postulate 5
Through any two points there
exists exactly one line.
Postulate 6
A line contains
at least two points.
Postulate 7
If two lines intersect, then
their intersection is exactly
one point.
Postulate 8
Through any three noncollinear
points there exists exactly one
plane.
B
A
T
C
Postulate 9
A plane contains at least three
noncollinear points.
Postulate 10
If two points lie in a plane, then
the line containing them lies in
the plane.
Postulate 11
If 2 planes intersect, then their
a line
intersection is ___________.
Workbook
Page 23 (1-5)
Lesson 2.2
Biconditional
Statements
You will learn to…
* recognize and use definitions
* recognize and use biconditional
statements
All definitions can be
interpreted “forward” and
“backward.”
All definitions are
biconditional.
For example,
perpendicular lines
are defined as
two lines that intersect
to form one right angle.
If two lines are perpendicular,
then they intersect to form
one right angle.
If two lines intersect to form
one right angle, then they are
perpendicular.
A biconditional statement
contains the phrase
“if and only if.”
Two lines are perpendicular
if and only if they intersect
to form one right angle.
A biconditional statement
is true when the
original if-then statement
AND
its converse are both true.
1. Two angles are supplementary
if and only if the sum of their
measures is 180°.
if-then statement:
If two angles are supplementary,
then the sum of their measures is 180°.
converse:
If the sum of the measures of two
angles is 180°, then they are
supplementary.
2. If an angle is 135˚, then it is
an obtuse angle.
converse:
If an angle is obtuse, then its
measure is 135°.
Can we write a biconditional
statement?
counterexample?
3.
If two angle measures add up to 90˚,
then they are complementary angles.
converse:
If two angles are complementary, then
the sum of their measures is 90°.
Can we write a biconditional
statement?
Two angles are complementary if and
only if the sum of their measures is 90°.
Workbook
Page 25 (1-7)
Lesson 2.3
Deductive
Reasoning
You will learn to…
* use symbolic notation to represent
logical statements
* form conclusions by applying laws
of logic
Using these phrases, write the
conditional statement.
p: mB = 90˚ q: B is a right angle
1. p  q
2. q  p
3. ~ p  ~ q
4. ~ q  ~ p
5. p  q
If mB = 90˚,
If  B is a right
then

B
is
a
Ifangle,
mB then
≠ 90˚,
right
angle.
If mB
 Bis=
a
then
Bnot
is not
90˚
=angle.
90˚
if
right
angle,
then
amB
right
and
only
if  B
mB
≠ 90˚
is a right angle.
Deductive Reasoning
uses facts to make a
logical argument.
definitions, properties,
postulates, theorems, and
laws of logic
Law of Detachment
Given
facts
Therefore:
pq
p
q
hypothesis
is true
conclusion
must be true
You can use these symbols when
asked to explain your reasoning.
Law of Detachment
q
p
If I learn my facts, then I will
pass geometry.
p
I learned my facts.
q
Therefore, I passed geometry.
Law of Syllogism
p

q
Given
qr
facts
Therefore: p  r
You can use these symbols when
asked to explain your reasoning.
Law of Syllogism
p
q
If I pass geometry, then my
dad will be happy.
q
If my dad is happy, then I will
r
get a cell phone.
p
Therefore, if I pass geometry,
then I will get a cell phone.
r
6. Is this argument valid?
If 2 lines in a plane are
parallel, then they do
pq
not intersect.
Coplanar lines n and m
are parallel.
p
Therefore, lines n and
q
m do not intersect.
VALID – Law of Detachment
7. Is this argument valid?
If 2 angles are supplementary, then
the sum of their measures is 180˚.
If 2 angles form a linear
pair, then they are
supplementary.
 pq
rp 
pr  qp
Therefore, if 2 angles form
r

q
a linear pair, then the sum
of their measures is 180˚
VALID – Law of Syllogism
8. Is this argument valid?
If 2 angles are a linear pair, then
the sum of their measures is 180˚.
m1 + m2 = 180˚
Therefore, 1 and 2
are a linear pair.
INVALID
pq
q
p
9. Is this argument valid?
If you live in Canada, then you live
in North America.
rp 
 qq
If you live in South
Carolina, then you live in
r
 qq
p
North America.
Therefore, if you live in
pr
Canada, then you live in
South Carolina
INVALID
If you use this product,
then you will have even-toned skin.
If you
have
even-toned
skin,
If you
use
this product,
youyou
will will
be beautiful.
then
be beautiful.
then
Lesson 2.4
Properties of Equality
and Congruence
You will learn to…
* use properties from algebra
* use properties of length and
measure to justify segment
and angle relationships
Equality Properties
Reflexive Property
Symmetric Property
Transitive Property
Addition Property
Subtraction Property
Multiplication Property
Division Property
Distributive Property
Substitution Property
Reflexive Property
XY  XY
m X  m X
Symmetric Property
If MN  20,
then 20  MN
If mN  mM
then mM  mN
Transitive Property
If XY = ST
and ST = 10,
then XY = 10
If mA = mB
and mB = 10°,
then mA = 10°
Division Property
If 8x=16,
then x=2.
Addition Property
If x-7=5,
then x=12.
Multiplication Property
If ½ x = 7,
then x=14.
Subtraction Property
If x+3=7,
then x=4.
Substitution Property
If
2
A=x
and x=6,
then A=36.
If 4 + 7x – 10 = 24,
Then 7x - 6 = 24
Distributive Property
If B=2(4x + 3),
then B=8x + 6.
If 4x + 7x = 24,
Then 11x = 24
Proofs !!
Memorize definitions, postulates,
and theorems as we learn them.
Write out entire proof each time one
is in the assignment.
Don’t give up!!!! You can do it!!!!
Let’s Practice…
2.
4+2(3x+5)=11-x Given
4+6x+10=11-x
Distributive prop.
14+6x =11-x
14 + 7x = 11
Substitution
Addition prop.
7x = - 3
x = - 3/7
Subtraction prop.
Division Prop.
4.
1/
5
x + 4 = 2x + 3/5 Given
1x + 20 = 10x + 3 Multiplication Prop
20 = 9x + 3
Subtraction Prop
Subtraction Prop
17 = 9x
17/
9
=X
Division Prop
5. Given that MN-PQ,
show that MP=NQ
MN = PQ
MP = MN + NP
MP = PQ + NP
Q
P
N
Given
Segment Addition
Postulate
NQ = PQ + NP
Substitution Prop
Segment Addition
Postulate
MP = NQ
Substitution Prop
M
7. Given mAQB=mCQD,show
that mAQC=mBQD Q
mAQB = mCQD
Given
D
A
B
C
mAQB + mBQC = mAQC Angle Addition
Postulate
mCQD + mBQC = mAQC Substitution
Angle Addition
mCQD + mBQC = mBQD
Postulate
mAQC = mBQD
Substitution
8. Given mRPS=mTPV
and mTPV=mSPT
P
show that mRPV=3(mRPS)
mRPS = mTPV
mTPV = mSPT
Given
R
S
T
V
Given
mRPS = mSPT
Transitive Prop
Angle Addition
mRPV=
mRPS+mSPT+mTPV Postulate
mRPV=
Substitution
mRPS+mRPS+mRPS
mRPV = 3(mRPS)
Distributive Prop
You can use definitions
as reasons in proofs.
Statements
Reasons
1)  A is a right
angle
1) Given
2) m A = 90˚
2) Def. of right angles
Statements
Reasons
1) m A = 90˚
1) Given
2)  A is a right
angle
2) Def. of right angles
C
1
Statements
A
Reasons
D
B
1) AB  CD
1) Given
2)  1 is a
right angle
2) Def. of  lines
C
1
A
D
B
Statements
1)  1 is a
right angle
Reasons
1) Given
2) AB  CD
2) Def. of  lines
1
2
Statements
Reasons
1) 1 and 2 1) Def. of vertical
angles
are vertical
angles
2
1
Statements
Reasons
1) 1 and 2 1) Def. of linear pair
are a linear
pair
Statements
1) AB = CD
2) AB  CD
Reasons
1) Given
2) Def. of 
Statements
1) AB  CD
Reasons
1) Given
2) AB = CD 2) Def. of 
Statements
Reasons
1) m1 = m2 1) Given
2)  1   2
2) Def. of 
Statements
1)  1   2
Reasons
1) Given
2) m1 = m2 2) Def. of 
Lesson 2.5
Proving Statements
about Segments
You will learn to…
* justify statements about
congruent segments
* write reasons for steps in a proof
use practice sheet of proofs
Proofs !!
Memorize definitions, postulates,
and theorems as we learn them.
Write out entire proof each time one
is in the assignment.
Don’t give up!!!! You can do it!!!!
Reflexive Property
of Congruence
XY  XY
Symmetric Property
of Congruence
If XY  JK,
then JK  XY
Transitive Property
of Congruence
If XY  JK
and JK  MN ,
then XY  MN
1. Given: EF = GH
Prove: EG  FH
H
G
E
F
(Proof is on next slide)
1.
Statements
1) EF = GH
2) EF + FG = GH + FG
3) EG = EF + FG
FH = GH + FG
4) EG = FH
5) EG  FH
Reasons
1) Given
2) Addition Prop.
Segment Addition
3)
Postulate
4) Substitution
5) Def. of 
2. Given: RT  WY, ST = WX
Prove: RS  XY
T
S
R
X
W
Y
2.
Statements
Reasons
1) Given
1) RT  WY
2) RT = WY
2) Def. of 
3) RT = RS + ST
3)Segment Addition
WY = WX + XY
Postulate
4) RS + ST = WX + XY 4) Substitution
5) ST = WX
5) Given
6) RS + ST = ST + XY 6) Substitution
7) RS = XY
7) Subtraction Prop.
8) Def. of 
8) RS  XY
3. Given: X is the midpoint of MN
and MX = RX
Prove: XN = RX
M
S
X
R
N
3.
Statements
1) X is the
midpoint of MN
2) NX = MX
3) MX = RX
4) NX = RX
Reasons
1) Given
2) Def. of midpoint
3) Given
4) Transitive Prop.
Paragraph proof example for #1
Since EF = GH, EF + FG = GH + FG by
the Addition Property. EG = EF + FG
and FH = GH + FG by the Segment
Addition Postulate. By Substitution,
EG = FH. Therefore, EG  FH by the
definition of congruent segments.
Paragraph proof example for #3
So, I was chillin’ with the homeboys and
my homeboy Sherrod tells me, “Dave, x is
the midpoint of MN, so NX = MX.” I said,
“Sherrod, how do you figure?” Sherrod
tells me “The definition of midpoint says
so!” So I was like, “yo, Sherrod, did you
know MX = RX,” and he said, “really, well
then NX = RX Dawg. “Sherrod, my homie,
I didn’t know you were so smart,” I said,
“how did you figure that out?” He was
like, “Substitution, my brother!”
David Mathews
# 17
Statements
1) XY = 8, XZ = 8,
2) XY = XZ
3) XY  XZ
4) XY  ZY
5) XZ  ZY
# 18
Statements
1) NK  NL, NK = 13
2) NK = NL
3) NL = 13
Lesson 2.6
Proving Statements
about Angles
You will learn to…
* use angle congruence
* prove properties about special
pairs of angles
Theorem 2.3
Right Angle Congruence
Theorem
right angles are
All ________
congruent
__________.
A is supplementary to 40°
B is supplementary to 40°
What do you know about
A and B?
A  B
Theorem 2.4
Congruent Supplements
Theorem
If 2 angles are
supplementary to the same
angle, then they are
congruent
_______________.
Using the Congruent
Supplements Theorem…
Statements
1) 1 & 2 are supp.
1 & 3 are supp.
Reasons
2)  2   3
2) Congruent
Supplements
Theorem
A is complementary to 50°
B is complementary to 50°
What do you know about
A and B?
A  B
Theorem 2.5
Congruent Complements
Theorem
If 2 angles are
complementary to the same
angle, then they are
congruent
_______________.
Using the Congruent
Complements Theorem…
Reasons
Statements
1) 1 & 2 are comp.
1 & 3 are comp.
2)  2   3
2) Congruent
Compliments
Theorem
Postulate 12
Linear Pair Postulate
If two angles form a
linear pair, then they are
supplementary
_______________.
Using the Linear Pair
Postulate…
Statements
1) 1 & 2 are a
linear pair
Reasons
1) Def. of linear
pair
2) 1 & 2 are
supplementary
2) Linear Pair
Postulate
3) m1 + m2 = 180 3) Def. of
supplementary
Theorem 2.6
Vertical Angles Theorem
Vertical angles are
congruent
_______________.
Using the Vertical Angles
Theorem…
Statements
1) 1 & 2 are
vertical angles
Reasons
1) Def. of vertical
angles
2) 1  2
2) Vertical Angles
Theorem
1. Given: 1  2 , 3  4 ,
2  3
Prove: 1  4
1
2
4
3
1.
Statements
1. 1  2 ,
2  3
2. 1  3
3. 3  4
4. 1  4
Reasons
1. Given
2. Transitive Prop.
3. Given
4. Transitive Prop.
2. Given: m1 = 63˚,1  3 ,
3  4
Prove: m4 = 63˚
1 2
3
4
2.
Statements
1. m1 = 63˚,
1  3 , 3  4
2. 1  4
3. m1 = m4
4. m4 = 63˚
Reasons
1. Given
2. Transitive Prop.
3. Def of 
4. Substitution
3. Given: DAB & ABC are right
angles , ABC  BCD
Prove: DAB  BCD
D
C
A
B
3.
Statements
1. DAB & ABC
are right angles
2. DAB  ABC
3. ABC  BCD
4. DAB  BCD
Reasons
1. Given
2. All right s are 
3. Given
4. Transitive Prop.
4. Given: m1 = 24˚,m3 = 24˚
1 & 2 are complementary
3 & 4 are complementary
Prove: 2  4
1
2
4
3
4.
Statements
m1 = 24˚, m3 = 24˚
1. 1 & 2 are comp.
3 & 4 are comp.
2. m1 = m3
3. 1  3
4. 2  4
Reasons
1. Given
2. Substitution
3. Def of 
4. Congruent
Complements
Theorem
5.
Statements
Reasons
1. 1 and 2 are
a linear pair
1. Given
2 and 3 are
a linear pair
2. 1 and 2 are supp.
Linear Pair
2.
2 and 3 are supp.
Postulate
3. 1  3
3. Congruent
Supplements
Theorem
6.
Statements
1. QVW and RWV
Reasons
1. Given
are supplementary
Def. of
Linear Pair
a linear pair
3. QVW and QVP are 3. Linear Pair
Postulate
supplementary
4. QVP  RWV
4. Congruent
2. QVW and QVP are 2.
Supplements
Theorem
#24 & #26 for homework
#24
Statements
1)  3 and  2 are complementary
2) m1 + m2 = 90
3) m 3 + m2 = 90
4) m1 + m2 = m3 + m2
5) m1 = m3
6) 1  3
#26
Statements
1) 4 and 5 are vertical angles
2) 6 and 7 are vertical angles
3) 4  5 , 6  7
4) 5  6
5) 4  7