Transcript Chapter 2 PowerPoint Slides File

Chapter 2
Section 2.1 – Conditional
Statements
Objectives:
To recognize conditional statements
To write converses of conditional statements
• Conditional  another name for an “if-then”
statement.
▫ Ex: If you do your homework, then you will pass
this class.
• Every conditional has two parts:
▫ 1. Hypothesis  part following the “if”
▫ 2. Conclusion  part following the “then”
• Ex: Identify the hypothesis and conclusion in
each statement.
If the Angels won the 2002 World Series,
then the Angels were world champions in 2002.
If x – 38 = 3, then x = 41
If Kobe Bryant has the basketball, then he will
shoot the ball everytime.
• Ex: Write each sentence as a conditional
statement.
A rectangle has four right angles.
A tiger is an animal.
An integer that ends in 0 is divisible by 5.
• Every conditional statement will have a truth
value associated with it: either true or false.
▫ A conditional is true if every time the hypothesis
is true, the conclusion is also true.
▫ A conditional is false if a counter-example can be
found for which the hypothesis is true but the
conclusion is false.
• Ex: Show each conditional to be false by finding
a counter-example.
If it is February, then there are only 28 days
in the month.
If the name of a state contains the word New,
then it borders the ocean.
• Converse  occurs when the hypothesis and
conclusion of a conditional statement are
switched.
• Ex:
Conditional
If two lines intersect to form right angles, then
they are perpendicular.
Converse
If two lines are perpendicular, then they intersect
to form right angles.
• It is important to see that just because the original
conditional was true, does not mean the converse
will also be true.
Take the following for example:
Conditional
If a figure is a square, then it has four sides.
True
Converse
If a figure has four sides, then it is a square.
False
• Summary – Conditional Statements/Converses
Statement
Example
Symbolic
Form
You Read
It
Conditional
If an angle is a straight angle,
then its measure is 180
degrees.
pq
If p, then q
Converse
If the measure of an angle is
180 degrees, then it is a
straight angle.
qp
If q, then p
•Homework #8
•Due
•Page 83 – 84
▫# 1 – 17 odd
▫# 23 – 31 odd
Section 2.2 – Biconditionals and
Definitions
• Objectives:
To write biconditionals
To recognize good definitions
• Biconditional  the statement created when a
conditional and its converse are combined into a
single statement with the phrase “if and only if”
▫ This can only be done if both the conditional and the
converse are true.
• Ex: Take each conditional and write its converse.
If both are true, then write a biconditional.
If two angles have the same measure, then the
angles are congruent.
If three points are collinear, then they lie on
the same line.
• Summary – Biconditional Statements
p
A biconditional combines p  q and q  p as
q.
Statement
Example
Symbolic
Form
Biconditional
An angle is a
p
straight angle if and
only if its measure is
180 degrees.
q
You Read It
p if and only if q
• Good Definition  a statement that can help you
identify or classify an object. A good definition
has three important components.
▫ 1. A good definition uses clearly understood terms.
The terms should be commonly understood or
already defined.
▫ 2. A good definition is precise. They will avoid
such words as large, sort of, and almost.
▫ 3. A good definition is reversible. That means that
you can write a good definition as a true
biconditional.
•Homework #9
•Due
•Page 90
▫# 1 – 23 odd
Section 2.3 – Deductive Reasoning
• Objectives:
To use the Law of Detachment
To use the Law of Syllogism
• Deductive Reasoning (Logical Reasoning)  the
process of reasoning logically from given
statements to a conclusion. If the given
statements are true, deductive reasoning will
produce a true conclusion.
Examples of Deductive Reasoning?
• Property – Law of Detachment
▫ If a conditional is true and its hypothesis is true,
then its conclusion is true.
▫ Symbolic form:
If p  q is a true statement and p is true, then q
is true.
• Ex: What can be concluded about each given true
statements?
If M is the midpoint of a segment, then it
divides the segment into two congruent segments.
M is the midpoint of AB.
If a pitcher throws a complete game, then he
should not pitch the next day. Jered Weaver is a
pitcher who has just pitched a complete game.
• Property – Law of Syllogism
▫ If p  q and q  r are true statements, then p  r
is a true statement.
• The Law of Syllogism allows us to state a
conclusion from two true conditional statement
when the conclusion of one statement is the
hypothesis of the other statement.
• Ex: Use the Law of Syllogism to draw a conclusion
from the following true statements.
If a number is prime, then it does not have
repeated factors.
If a number does not have repeated factors,
then it is not a perfect square.
If a number ends in 6, then it is divisible by 2.
If a number ends in 4, then it is divisible by 2.
•Homework #10
•Due
•Page 96 – 97
▫# 1 – 21 odd
Section 2.4 – Reasoning in Algebra
• Objectives:
To connect reasoning in algebra and
geometry.
• Summary – Properties of Equality
Addition Property
If a = b, then a + c = b + c
Subtraction Property
If a = b, then a – c = b – c
Multiplication Property
If a = b, then ac = bc
Division Property
𝑎
𝑐
If a = b and c ≠ 0, then =
𝑏
𝑐
Reflexive Property
a=a
Symmetric Property
If a = b, then b = a
Transitive Property
If a = b and b = c, then a = c
Substitution Property
If a = b, then b can replace a in
any expression
• Summary – Properties of Congruence
Property
Example
Reflexive Property
AB ≈ AB
<A ≈ <A
Symmetric Property
If AB ≈ CD, then CD ≈ AB
If <A ≈ <B, then <B ≈ <A
Transitive Property
If AB ≈ CD and CD ≈ EF, then AB ≈ EF
If <A ≈ <B and <B ≈ <C, then <A ≈ <C
Section 2.5 – Proving Angles Congruent
• Objectives:
To prove and apply theorems about angles
• Theorem  a statement proved true by deductive
reasoning through a set of steps called a proof.
• In the proof of a theorem, a “Given” list shows
you what you know from the hypothesis of the
theorem. The “givens” are then used to prove the
conclusion of a theorem.
• Theorem 2.1 – Vertical Angles Theorem
▫ All vertical angles are congruent.
1
3
4
2
<1 ≈ <2 and <3 ≈ <4
• Theorem 2.2 – Congruent Supplements Theorem
▫ If two angles are supplements of the same angle (or
of congruent angles), then the two angles are
congruent.
• Theorem 2.3 – Congruent Complements Theorem
▫ If two angles are complements of the same angle (or
of congruent angles), then the two angles are
congruent.
• Theorem 2.4
▫ All right angles are congruent.
• Theorem 2.5
▫ If two angles are congruent and supplementary,
then each is a right angle.
• Ex: Solve for x and y.
y°
3x°
75°
•Homework #11
•Due
•Page 112 – 113
▫# 1 – 6 all
▫# 8 – 18 all