2.1 Conditional Statements
Note, there will be a notes
handout for 2.1 given in class.
A Conditional Statement has TWO
parts, the HYPOTHESIS and
CONCLUSION
In if-then form
If -> HYPOTHESIS p
Then -> CONCLUSION  q
(it’s not necessarily these letters all
the time)
Writing a statement in if-then form.
People who go to USC need help.
If you go to USC, then you need help
A Troy student is a Warrior.
If you are a Troy student, then you are a Warrior.
Write the statement in if-them form.
All mountains have rocks.
If it is a mountain, then it has rocks.
A lover of potatoes is a lover of French fries.
If you love potatoes, then you love French fries.
It just takes one COUNTEREXAMPLE to
prove something is wrong.
If you go to USC, then you are smart.
False, virtually every USC student
If x2 = 25, then x = 5
False, x = -5
All triangles are equilateral
False, some triangles have different side lengths.
All even numbers can be divided by 2.
True
A CONVERSE is when you reverse
the original conditional statement.
If you are in room 302, then the room is
cold.
If it is cold, then you are in room 302.
The converse is NOT always true!!!!
Write the converse and say whether it is
true, or state a counterexample.
If a student is in room 302, the student is
in math class.
If a student is in math class, the student
is in room 302.
Mr. Booze’s class.
A statement can be changed by negation,
which is writing the negative of a statement.
Conditional If you are in room 302, you are in Mr.
pq
Converse
q p
Inverse
~ p ~ q
Kim’s class.
If you are in Mr. Kim’s class, then you
are in room 302.
If you are not in room 302, you are not in
Mr. Kim’s class.
Negation symbol
Contrapositive
~ q  ~ p If you are not in Mr. Kim’s class, you are
not in room 302.
• Conditional  If you go to Troy, then you
are a student.
• Converse  If you are a student, then you
go to Troy.
• Inverse If you don’t go to Troy, then
you’re not a student.
• Contrapositive If you are not a student,
then you don’t go to Troy.
THE Conditional statement and the
contrapositive are EQUIVALENT (if one is
true the other is true, if one is false, the
other is false.
THE CONVERSE and INVERSE are
EQUIVALENT (if one is true the other is
true, if one is false, the other is false.
Postulate 5: Through any Postulate 6: A line
two points there is
contains at least two
EXACTLY one line
points.
Postulate 7: If two lines intersect, then they intersect
in exactly one point.
Postulate 8: Through any
three noncollinear points
there is EXACTLY one
plane.
Postulate 9: A plane
contains at least three
noncollinear points.
Postulate 10: If two
points are in a plane,
then the line that
contains the points is in
that plane.
If not, it’d be like this.
Postulate 11: If two
planes intersect, then
their intersection is a
line.
P
Name two lines on
plane AZHU that
are not drawn.
T
AH and ZU
A
Z
H
I
U
How do you know line
Through points A, U, T,
IZ is on plane IPZA ?
One plane
there is exactly ________
If two points are on a
plane, the line
containing them is on
the plane. (Post)
Postulate ____
8
due to __________
2.2 – Definitions and
Biconditional Statements
Definition of Perpendicular
lines (IMPORTANT): Two
lines that intersect to form
RIGHT ANGLES!
 perpendicular symbol
A line perpendicular to a
plane is a line that intersects
the plane in a point that is
perpendicular to every line in
the plane that intersects it.
All definitions work forwards and backwards
If two lines are perpendicular, then they form a right angle.
If two lines intersect to form right angles, then they are
perpendicular.
All definitions work forwards and backwards
If two lines are perpendicular, then they form a right angle.
If two lines intersect to form right angles, then they are
perpendicular.
If a conditional statement and its converse are both true, it is
called biconditional, and you can combine them into a “if and
only if” statement
Two intersecting lines are perpendicular if and only if they
form right angles.
Z
W
T
True or false? Why? (Check some hw)
Y
 WVT and  YVX are
X
complementary.
V
S
U
R
 WVZ andRVS form a linear pair.
 YVU and TVR are supplementary
Y, V, and S are collinear
Write the conditional statement and the
converse as a biconditional and see if
it’s true.
If two segments are congruent, then
their lengths are the same.
If the lengths of the segments are the
same, then they are congruent.
Two segments are congruent if and only
if their lengths are the same.
TRUE!
Write the conditional statement and the
converse as a biconditional and see if
it’s true.
If B is between A and C, then AB + BC =
AC
If AB + BC = AC, then B is between A and
C
B is between A and C if and only if AB +
BC = AC
TRUE!
Write the converse of the statement, then write the
biconditional statement. Then see if the biconditional
statement is true or false. (Check more hw)
If x = 3, then x2 = 9
If x2 = 9, then x = 3
x = 3 if and only if x2 = 9
False, x = -3 is a counterexample
If two angles are a linear pair, then they are
supplementary angles.
If two angles are supplementary, then they form a linear
pair
Two angles are a linear pair if and only if they are
supplementary.
False, they don’t have to be on the same line.
Split up the biconditional into a conditional statement
and its converse.
Pizza is healthy if and only if it has bacon.
Students are good citizens if and only if they follow the
ESLRs.
2.4 – Reasoning with
Properties from Algebra
Addition Prop. =
If a  b and c  d , then a  c  b  d
Subtraction Prop. = If a  b and c  d , then a  c  b  d
Multiplication Prop.= If a  b, then ca  cb
Division Prop. =
a b
If a  b and c  0, then 
c c
Substitution Prop. = If a  b, then either a or b
may be substituted for the other
in any equation (or inequality )
Reflexive Prop. =
aa
Symmetric Prop. =
If a  b, then b  a
Transitive Prop. =
If a  b and b  c , then a  c
Reasons
5 x  12  13
5 x  25
x5
Given Equation
Addition Prop =
Division Prop =
Reasons
1
x25
2
x  4  10
x6
Given Equation
Multiplication Prop =
Subtraction Prop =
Reflexive Prop. Of
equality
DE  DE
mD  mD
Symmetric Prop.If DE  FG , then FG  DE
Of equality
If mD  mE , then mE  mD
Transitive Prop. If DE  FG and FG  JK ,
Of equality
then DE  JK
If mD  mE and mE  mF ,
then mD  mF
We will fill in the blanks
Given : MA  TH
M
1)
A
MA  TH
T
H
Prove : MT  AH
1) Given
2)
2)
3)
3)
4)
4)
5)
5)
U
D
C
1
2
1) m2  30,
Given : m2  30, mUDK  50
Prove : m1  20
K
1) Given
mUDK  50
2)
2)
3)
3)
4)
4)
5)
5)
A
L
N
G
E
Given : AN  ES , NG  LE
S
P rove : AG  LS
2.6 – Proving Statements
about Angles
Copy a segment
1) Draw a line
Use same radius
for both circles, so
segments are
congruent.
Center of
compass
Writing
end
GREEN
DOT
DRAWN
PART
(compass
and ruler)
2) Choose point on line
3) Set compass to original radius, transfer it to new line,
draw an arc, label the intersection.
__________ Property
A  A
AB  AB
Symmetric Property
If A  B, then ____  ______
If AB  CD, then _____  ______
_________ Property
If A  B and B  C , then _____  ______
If AB  CD and CD  EF , then _____  ______
Right Angle Congruence Thrm - All ______ angles are _______
Congruent Supplements Theorem
If two angles are ____________ to the same angle (or to congruent angles),
then they are congruent.
If _____ + _____ = 180 and _____ + ____ = 180, then ____
 ____
Congruent Complements Theorem
If two angles are ____________ to the same angle (or to congruent angles),
then they are congruent.
If _____ + _____ = 90 and _____ + ____ = 90, then ____
 ____
Vertical Angles Thrm - _____ angles are ______
Linear Pair Postulate – If two angles form a linear pair, then they are
_________
L
W
1 and 2 are supplement ary.
R
3 and 4 are supplement ary.
I
O
A
2  3.
m1  55
Find
S
Z
m2 
m3 
m4 
IAO and OAZ are complement ary.
mIAO  35
Find
mWAI 
mRAI 
mRAL 
mIAS 
Given
E
T
1
R
Prove
2
A
mERA  90
Given
mERA  90
1 and 2 are
complementary
2
1
3
1, 2 are lin. pair
2, 3 are lin. pair
1, 2 are lin. pair
Given
2, 3 are lin. pair
Prove m1  m3
Given
Given m5  m7
P
J
5 6
K
N
7
M
m5  m7
mJKN  mPKM
2.5-9 Number 2
Prove mJKN  mPKM
Given
Given mPQT  mVQR
T
V
P
9
8
R
Prove m8  m10
10
Q
mPQT  mVQR
m8  m10
Given