Download Negation-writing the negative of the statement

2.1
CONDITIONAL STATEMENTS
Conditional Statement- a logical statement that has two parts. It’s
sometimes written in an “if-then” statement.
Hypothesis/Antecedent –
Conclusion/ConsequentEx.1: Write as a conditional statement:
All mammals breathe oxygen.
Converse- switching the hypothesis and conclusion (switch order)
Ex. 2: Write a counterexample and the converse:
If a number is odd, then it is divisible by 3.
Negation-writing the negative of the statement (INSERT “NOT”)
Ex. 3: mA  45
Inverse- negate the hypothesis and conclusion (keep the same order but
take the negation)
Contrapositive- negate the hypothesis and conclusion of the converse
(switch order and take the negation)
Ex. 4: Write the inverse, converse, and contrapositive of the statement.
“If the amount of available food increases, the deer population increases”.
Equivalent Statements-
Law of Contrapositives:
POINT-LINE-PLANE POSTULATE:

Through any two points there is exactly 1 line

A line consists of at least 2 points

If 2 lines intersect, their intersection is a POINT

Through any 3 noncollinear pts, there is a plane

A plane consists of at least 3 noncollinear points

If 2 pts are in a plane, then the line containing them are in the same
plane

If 2 planes intersect, their intersection is a LINE
Ex. 5: Decide whether the statement is true or false. If it’s false, give a
counterexample.
a) “Three points are always contained in a line”.
b) A line can contain more than two points.
2.2 DEFINITIONS & BICONDITIONAL STATEMENTS
Def. Of Perpendicular (  ) lines:
Def. of a line  to plane:
Every definition can be written as two if-then statements. The original and
its converse.
Biconditional Statement: If and only if (IFF)
A biconditional statement can be true or false. In order for it to be true,
both the conditional and its converse must be true.
Ex.1: Rewrite the biconditional statement as a conditional statement and its
converse: Two lines intersect if and only if their intersection is exactly one
point.
Ex. 2: Write the conditional and the converse. x 2  49 iff x  7. Is it a
true biconditional?
Ex.3: The following statement is true. Write the converse and decide
whether it is true or false. If the converse is true, combine it with the
original to form a biconditional. If x 2  4, then x  2 .
2.3 DEDUCTIVE REASONING
Symbol notation:
pq
is the same as if p, then q
is the same as p implies q
Converse:
q p
Inverse:
p
q
Contrapositive:
q
p
Biconditional:

Ex. 1: P= a quadrilateral has four right angles
Q= the quadrilateral is a square
Ex. 2: P= Today is Monday
Q=There is no school
INDUCTIVE REASONING- previous examples and patterns are used to
form a conjecture.
DEDUCTIVE REASONING- uses facts, properties and definitions in a
logical order to write a logical argument.
Two laws of deductive reasoning: Law of Detachment, Law of Syllogism
LAW OF DETACHMENT-
Ex. 3: State whether the argument is valid.
a) Mike knows that if he does not do his chores in the morning, he will
not be allowed to play video games later that day. Mike does not play
video games on Friday afternoon. So Mike did not do his chores on
Friday morning.
b) If two angles are vertical, then they are congruent. ABC and
DBE are vertical. Conclusion: DBE and ABC are congruent.
c) Sarah know that all sophomores take driver education in her school.
Hank takes driver education. So Hank is a sophomore.
LAW OF SYLLOGISM-
Ex 4: Write a conditional statement that can be written from the following
statements.
a) If it’s Wednesday, then its late start. If it’s late start, then school
starts at 9:40.
b) If a fish can swim at 110 km/h, then it’s a sailfish. If a fish swims at
68 mi/h, then it swims at 110 km/h.
2.4 REASONING WITH PROPERTIES FROM ALGEBRA
ALGEBRAIC PROPERTIES OF EQUALITY (used in equations)
Addition Prop of Equality:
if a = b, then
a+ c=b+ c
Subtraction Prop of equality:
if a = b, then
a– c=b– c
Multiplication Prop of Equality:
if a = b, then
ac = bc
a b
=
Division Prop of Equality:
if a = b, then
c0
c c
Reflexive Prop of Equality:
Symmetric Prop of Equality:
Transitive Prop of Equality:
a=a
if a = b, then
if a = b, & b = c, then
Substitution Property
if a = b, then
value of “b” in place of “a”
b = a
a = c
you can substitute the
Solve for the variable and write a reason for each step:
Ex. 1: 3x + 12 = 8x – 18
Ex. 2: -2(-w+3) = 15
GEOMETRIC PROPERTIES OF EQUALITY
Segment Length
Refl Prop of Equality:
Symm Prop of Equality:
AB = AB
if AB = CD, then CD = AB
Trans Prop of Equality:
if AB = CD & CD = EF,
then AB = EF
Angle Measure
mA=mA
if mA = mB,
then mB = mA,
If mA = mB,
& mB = mC ,
then mA =mC
Ex. 3:
AC = BD. Verify that AB = CD
Ex. 4: A baseball diamond is shown below. The pitcher’s mound is at
Use the information to find m 4 :
m1  m2  m3  180
m1  m2  93
m3  m4  180
2.5
3 .
PROVING STATEMENTS ABOUT SEGMENTS
Theorem- true statement that follows other true statements
All thm’s MUST be PROVED.
To prove something is true----- Proof
To prove something is false----Counterexample
Properties of Congruence: (used with the congruent symbol)
Reflexive Property of Congruence:
For any segment AB, AB  AB
Symmetric Property of Congruence.
If AB  CD, thenCD  AB
Transitive Property of Congruency:
AB  CD, andCD  EF ,then AB  EF
Recall:
Definition of congruence:
DEF. OF CONGRUENT ANGLES:
2-column proof:
Ex.1:
Given: LK = 5; JK=5;
Prove: LK  JL
JL  JL
Ex. 2:
Given: Q is the midpoint of PR
Prove: PQ = ½ PR
Statements
1. Q is the midpoint of PR
2. PQ = QR
3. PQ + QR = PR
4. PQ + PQ = PR
5. 2(PQ) = PR
6. PQ = ½ PR
Reasons
Ex. 3:
Given: X is the midpoint of MN
Prove: RX=XN
Ex. 4:
Given: AB  BC ; BC  CD
Find BC
Constructing a congruent segment:
Draw a long segment using a straightedge
Label one endpoint on the new segment
Set your compass on the original segment and measure the distance using
the pencil
Place the compass on the endpoint you made and mark the distance from the
compass.
Label the new endpoint
Ex. 5:
2.6
PROVING STATEMENTS ABOUT ANGLES
Recall: Reflexive, Symmetric, Transitive Properties of Equality &
Congruency.
Ex.1: Given:
1  2, 3  4, 2  3
Prove: 1  4
Ex. 2: Given: m1  63 , 1  3, 3  4
Prove: m4  63
RIGHT ANGLE CONGRUENCE THM:
CONGRUENT SUPPLEMENTS THM:
CONGRUENT COMPLEMENTS THM:
LINEAR PAIR POSTULATE:
VERTICAL ANGLES THM: