Download 2.2 Day 1: Date: ______ Geometry A Conditional Statement is an

2.2 Day 1: __________________________________________
Geometry
Date: __________
A Conditional Statement is an __________
statement.
The __________________ is the part _____
following if.
The __________________ is the part _____
following then.
Ex 1). What are the hypothesis and the conclusion of the conditional statement?
a)
If a number is even, then it is divisible by 2.
Hypothesis: ________________________________________________________________________
Conclusion: ________________________________________________________________________
b)
If an angle measures 130, then the angle is obtuse.
Hypothesis: ________________________________________________________________________
Conclusion: ________________________________________________________________________
Ex 2). How can you write the following statement as a conditional?
a)
Adjacent angles share a side.
b)
Dolphins are mammals.
The __________ ____________ of a conditional statement is either true or false.
* To show the statement is true: show that every time the hypothesis is true, the conclusion is true.
* To show that a statement is false: find ________ ______ counterexample for which the hypothesis is
true, but the conclusion is false.
Ex 3). Is the conditional true or false? If it is false, find a counterexample.
a)
If you live in Copley, then you live in Ohio.
b)
If a number is divisible by 5, then it is odd.
c)
If a month 28 days, then it is February.
d)
If two angles form a linear pair, then they are supplementary.
Homework: pg. 99 #1, 3 – 13, 14 – 28(e)
2.2 Day 2: __________________________________________
Geometry
Date: __________
The ______________ of a statement p is the _____________ of the statement. The symbol ______ is
read “not p.”
Statement
Conditional
Converse
Inverse
Contrapositive
How to Write It
Example
Symbols
How to Read It
Truth Values:
Statement
Example
Truth Value
Conditional
Converse
Inverse
Contrapositive
Ex 1). What are the converse, inverse, and contrapositive of each of the conditional statements
below? What are the truth values of each?
a)
If a figure is a rectangle, then it is a parallelogram.
b)
If a vegetable is a carrot, then it contains beta carotene.
Homework: pg. 103 #1, 3 – 7, 11 – 14, 18 – 25
2.3: ________________________________________________
Geometry
Date: __________
A ________________________ is a single true statement that combines a true conditional and its true
converse. You can write a biconditional by joining the two parts of each conditional with the
phrase ________________________________.
Ex 1). What is the converse of the following true conditional? If the converse is also true, combine
the statements as a biconditional.
Conditional: If the sum of the measures of two angles is 90°, then the angles are complementary.
A biconditional combines 𝑝 → 𝑞 and 𝑞 → 𝑝 as _______________.
ex: A point is a midpoint if and only if it divides a
segment into two congruent segments.
Ex 2). What are the two conditional statements that form this biconditional?
Two numbers are additive inverses if and only if their sum is 0.
What makes a good definition?
1. A good definition uses _________ ____________________ terms.
2. A good definition is _______________.
3. A good definition is _______________. It can be written as a true _______________________.
Ex 3). Is this definition of an equilateral triangle reversible? If yes, write it as a true biconditional.
An equilateral triangle is a triangle with 3 congruent sides.
Ex 4). Which of the following is a good definition?
[A]
Dogs are animals with 4 paws.
[B]
Squares have 4 sides.
[C]
Tuesday is the day before Wednesday.
[D]
An acute angle has a small measure.
Homework: pg. 108 #1 – 3, 8 – 26(e), 45 – 50
2.4: _________________________________________________
Geometry
Date: __________
________________ _______________ is the process of reasoning _____________ from given statements
or facts to make a ___________________.
Law of _________________________:
If the hypothesis of a _______ conditional is _______, then
the conclusion is _______.
Ex 1). What can you conclude from the given true statement?
a) Given: If it is raining outside, then there are clouds in the sky. It is raining today.
b) Given: If Malcolm scores at least 85% on his final exam, then he will earn an A for the
term. Malcolm scores a 90% on his final exam.
c) Given: If two angles are vertical, then they are congruent. Angles 3 and 4 are vertical
angles.
Law of ________________________:
Symbols:
Examples:
Ex 2). What can you conclude from the given information?
a) Given: If a number is divisible by 12, then it is divisible by 6. If a number is divisible by 6,
then it is divisible by 3.
b) Given: If a figure is a square, then the figure is a parallelogram. If a figure is a rectangle,
then the figure is a parallelogram.
Ex 3). What can you conclude from the given information?
a) Given: If you live in Cincinnati, then you live in Ohio. If you live in Ohio, then you live in
the United States. Ken lives in Cincinnati.
b) If a river is more than 4000 mi long, then it is longer than the Amazon. If a river is longer
than the Amazon, then it is the longest river in the world. The Nile is 4132 mi long.
Homework: pg. 117 #1 – 3, 6 – 16(e), 19 – 24, 26, 28
2.5: ___________________________________________________
Geometry
Date: __________
Ex 1). What is the value of x?
Ex 2). Write a two-column proof:
a) Given: 𝟑(𝟐𝒙 − 𝟖) = 𝟔
Prove: 𝒙 = 𝟓
b) Given: 𝟒(𝒙 − 𝟕) − 𝟐(𝟑𝒙 + 𝟒) = 𝟐
Prove: 𝒙 = −𝟏𝟗
Important Properties:
REFLEXIVE PROPERTY
SYMMETRIC PROPERTY
TRANSITIVE PROPERTY
Ex 3). What is the name of the property of equality or congruence that justifies going from the first
statement to the second statement?
a) 7x + 3 = 24
b)
RS = ST and ST = TU
7x = 21
RS = TU
c) 5x = 10
10 = 5x
b)
3x – 4 = x and x = -2
3(-2) – 4 = -2
A _________ is a convincing argument that uses deductive reasoning. A proof _________________
shows why a conjecture is true. A _______________________ lists each statement on the left and the
reason (justification) on the right. Each statement must follow logically from the steps ___________
it.
Ex 4). Write a two-column proof for the following:
Given: 𝒈 = 𝟐𝒉
𝒈+𝒉 =𝒌
𝒌=𝒎
Prove: 𝒎 = 𝟑𝒉
Ex 5). Write a two-column proof:
Given:
𝟑𝒙+𝟓
𝟐
=𝟕
Prove: 𝒙 = 𝟑
Homework: pg. 124 #1 – 17
2.6: ________________________________________________
Geometry
Date: __________
A ______________ is a conjecture or statement that you prove _________
Important Theorems:
VERTICAL ANGLES THM
CONGRUENT
SUPPLEMENTS THM
THEOREM 2-4
THEOREM 2-5
Ex 1). What is the value of x?
a)
b)
Ex 2). Write a two column proof:
Given:
∠𝟐 ≅ ∠𝟑
Prove:
∠𝟏 ≅ ∠𝟐
CONGRUENT
COMPLEMENTS THM
Ex 3). Write a two-column proof.
Given:
𝒎∠𝟏 = 𝒎∠𝟑
Prove:
𝒎∠𝑨𝑬𝑪 = 𝒎∠𝑫𝑬𝑩
A
B
C
1
2
D
3
E
Ex 4). Write a two-column proof:
Given: ̅̅̅̅
𝑪𝑫 ≅ ̅̅̅̅
𝑬𝑭
̅̅̅̅
̅̅̅̅ ≅ 𝑭𝑮
𝑫𝑬
Prove: ̅̅̅̅
𝑪𝑬 ≅ ̅̅̅̅
𝑬𝑮
C
E
D
F
G
G
G
Ex 5). Write a two-column proof.
Given: ∠𝟒 𝒊𝒔 𝒂 𝒓𝒊𝒈𝒉𝒕 𝒂𝒏𝒈𝒍𝒆
∠𝟏 ≅ ∠𝟐
Prove: ∠𝟐 𝒂𝒏𝒅 ∠𝟔 𝒂𝒓𝒆
𝒄𝒐𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚
3
2
1
6
4
5
Homework: pg. 132 #1 – 3, 6 – 13, 16 – 17, 19