Transcript q - Methacton School District

Chapter 2
Midterm Review
By: Mary Zhuang, Amy Lu, Khushi
Doshi, Sayuri Padmanabhan, and
Madison Shuffler
Introduction
Write a two-column proof.
Given: 2(3x – 4) + 11 = x – 27
Prove: x = -6
Statement
Reason
2(3x – 4) + 11 = x – 27
Given
6x – 8 + 11 = x – 27
Distributive
6x + 3 = x – 27
Substitution
6x – x + 3 = x – x – 27
Subtraction
5x + 3 = -27
Substitution
5x + 3 – 3 = -27 – 3
Subtraction
5x = -30
Substitution
5x/5 = -30/5
Division
X = -6
Substitution
Euclid
Εὐκλείδης meaning, “good glory”
300 BC
Also know as Euclid of Alexandria
• Only a couple references that referred to him,
nothing much is known about him and his life.
• Known as the “father of geometry”
• Created a book called The Elements, one of the best
works for the history of mathematics
• The Elements serves as the main textbook for
mathematics, especially geometry. And that is where
“Euclid Geometry” came from, which is what we
learn today.
How does Euclid relate to Chapter 2?
Euclid actually created five postulates when he was alive, and we are
introduced to postulates in Chapter 2.
His five postulates are:
1. “A straight line segment can be drawn to join any two points”
(2.1 Postulate)
2. “Any straight line segment can be extended indefinitely in a
straight line.” (definition of line)
3. “Given any straight line segment, a circle can be drawn having
the segments as radius and one endpoints as center.”
4. “All right angles are congruent.” (right angle theorem)
5. “If two lines are drawn which intersect a third in such a way that
the sum of the inner angles on one side is less that two right
angles, then the two lines inevitable must intersect each other on
that side if extended far enough.” (parallel postulate)
2-1 Inductive Reasoning and
Conjectures
• Conjecture: An statement based on known
information that is believed to be true but not yet
proved
_______
• Inductive reasoning: Reasoning that uses a
number of specific examples or observations to
arrive at a plausible generalization
• Deductive reasoning: Reasoning that uses facts,
rules, definitions, and/or properties to arrive at a
conclusion
• Counterexample: Example used to prove that a
conjecture is ____
not true
2-1 Inductive Reasoning and
Conjectures
For example:
If we are given information on the quantity and
formation of the first 3 sections of stars, make a
conjecture on what the next section of stars would
be.
2-2 Logic
• Statement: sentence that must be either true or
false
- Statement n: We are in school
• Truth Value: whether the statement is true or
false
True
- Truth value of statement n is _______
• Compound Statement: two or more statements
joined:
- We are in school and we are in math class
• Negation: opposite meaning of a statement and
the truth value, it can be either true or false
not in school
- Negation of statement n is: We are ____
2-2 Logic
• Conjunction: compound statement using
“and”
- A conjunction is only true when all the statements in it are
_____
true
For example:
Iced tea is cold and the sky is blue – Truth value is _____
true
• Disjunction: compound statement using “or”
- A disjunction is true if at least one of the statements is true
For example:
May has 31 days or there are 320 days in an year – Truth
value is true
2-2 Logic
• Truth tables: organized method for truth value
of statements
Fill in the last column of each truth table:
Conjunction:
Disjunction:
p
q
p q
p
q
p q
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
2-2 Logic
• Venn diagram
- The center of the Venn diagram is the
conjunction, also called the “and” statement
- All the circles together make up the
disjunction, also called the “or” statement
Australia is the
conjunction
Continent
Australia
Island
Continent, Island, and
Australia is the
disjunction
2-3 Conditional Statements
• Conditional Statement: Statement that can be
written in if-then form
• Hypothesis: Phrase after the word “if”
• Conclusion: Phrase after the word “then”
_____
• Symbols: p → q, “if p, then q”, or “p implies q”
2-3 Conditional Statements
Symbols
Formed by
Example
hypothesis and
conclusion
they will cancel
school
Truth Value
Truth pTable
when
given Conditional
Statements:
→q
Using the given
If it snows, then True
Conditional
Converse
“switch”
q→p
Exchanging the
hypothesis and
conclusion
If they cancel
school, then it
snows
False
Inverse
“not”
∼p → ∼q
Replacing the
hypothesis and
conclusion with its
negation
If it does not
snow, then they
will not cancel
school
False
Contrapositive
“switch-not”
∼q → ∼p
Negating the
hypothesis and
conclusion and
switching them
If they do not
cancel school,
then it does not
snow
True
Biconditional
p
Joining the
conditional and
converse
It snows if and
only if they
cancel school
False
1
q
2-4 Deductive reasoning
• Law of Detachment: If p then q is true and p is
true then, q is true.
- Symbols: [(p→q) p]→ q
• Law of Syllogism: If p then q and q then r are
true, then p then r is also true.
- Symbols: [(p→q) (q→r)]→(p→r)
2-5 Postulates and Proofs
Postulate: a statement that describes a fundamental
relationship between basic terms of geometry
2.1 Through any __
2 points, there is exactly 1 line
2.2 Through any 3 points not on the _______
same line, there is
exactly 1 plane
2.3 A _____
line contains at least 2 points
2.4 A plane contains at least __
3 points not on the same line
2.5 If 2 points lie in a plane, then the entire _____
line containing
those points lies in that plane
point
2.6 If 2 lines intersect, then their intersection is a _____
2.7 If 2 _______
planes intersect, then their intersection is a line
2-5 Postulates and Proofs
• Theorem: A statement or conjecture shown to be
true
• Proof: A logical argument in which each
statement you make is supported by a statement
that is accepted as true
• Two-column proof: a formal proof that contains
statements and reasons organized in two
columns. Each step is called a statement and the
properties that justify each step are called
________
reasons
2-5 Postulates and Proofs
Steps to a good proof:
1.) List the given information
2.) Draw a diagram to illustrate the given
information (if possible)
3.) Use deductive reasoning
proved
4.) State what is to be ______
2-5 Postulates and Proofs
Definition of Congruent segments:
𝐴𝑀 = 𝑀𝐵 ↔ 𝐴𝑀 ≅ 𝐵𝑀
Definition of congruent Angles:
𝑚∠𝐴 = 𝑚∠𝐵 ↔ ∠𝐴 ≅ ∠𝐵
Midpoint Theorem:
midpoint of 𝐴𝐵, then 𝐴𝑀 ≅ 𝑀𝐵
If M is the _______
2-6 Algebraic Proofs
• The properties of equality can be used to justify each step
when solving an equation
• A group of algebraic steps used to solve problems form a
deductive argument
2-6 Algebraic Proofs
Given: 6x + 2(x – 1) = 30
Statements
1.) 6x + 2(x-1) = 30
2.) 6x + 2x – 2 = 30
3.) __________
8x – 2 = 30
4.) 8x – 2 + 2 = 30 + 2
5.) ________
8x = 32
6.) 8x/8 = 32/8
7.) x = 4
Prove: x = 4
Reasons
1.) ______
Given
2.) __________
Distributive
________
Property
3.) Substitution
4.) Addition Property
5.) Substitution
6.) Division Property
Substitution
7.) ____________
2-6 Algebraic Proofs
• Since geometry also uses variables, numbers, and
operations, many of the properties of equality used
in algebra are also true in geometry
2-7 Proving Segment Relationships
• Ruler Postulate: The points on any line can be
paired with real numbers so that given any
two points A and B on a line, A corresponds to
zero and B corresponds to a positive real
number. (This postulate establishes a number
line on any line)
• Segment Addition Postulate: 𝐵 is between 𝐴
and 𝐶 if and only if 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶
A
B
C
2-7 Proving Segment Relationships
Segment Congruence
• Reflexive Property: 𝐴𝐵 ≅ 𝐴𝐵
• Symmetric Property: If 𝐴𝐵 ≅ 𝐶𝐷, then 𝐶𝐷 ≅
𝐴𝐵
• Transitive Property: If 𝐴𝐵 ≅ 𝐶𝐷 and 𝐶𝐷 ≅
𝐸𝐹, then 𝐴𝐵 ≅ 𝐸𝐹
2-7 Proving Segment Relationships
For Example:
Given: A, B, C, and D are collinear, in that order; AB=CD
Prove: AC=BD
2-8 Proving Angle Relationships
• Addition Postulate (2.11): 𝑅 is in the interior of
∠𝑃𝑄𝑆 iff 𝑚∠𝑃𝑄𝑅 + 𝑚∠𝑅𝑄𝑆 = 𝑚∠𝑃𝑄𝑆
P
R
Q
S
2-8 Proving Angle Relationships
• 2.3 Supplement Theorem: if two angles form a
_______
linear pair, then they are _____________
supplementary
angles
• 2.4 Complement Theorem: If the noncommon
right
sides of two adjacent angles form a _____
angle, then the angles are _____________
complementary
angles
2-8 Proving Angle Relationships
• Theorem 2.5: Congruence of angles is reflexive,
symmetric, and transitive
Reflexive Property: ∠1 ≅ ∠1
• ________
• Symmetric Property: If ∠1 ≅ ∠2, then ∠2 ≅ ∠1
• ________
Transitive Property: If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then
∠1 ≅ ∠3
2-8 Proving Angle Relationships
• 2.6 Congruent Supplement Theorem: Angles
same angle or to
supplementary to the _____
congruent angles are _________
congruent
• If m∠1 + 𝑚∠2 = 180 and m∠2 + 𝑚∠3 = 180, then ∠1 ≅ ∠3
• 2.7 Congruent Complement Theorem: Angles
complementary
_____________ to the same angle or to
congruent angles are congruent
_________
• If m∠1 + 𝑚∠2 = 90 and m∠2 + 𝑚∠3 = 90, then ∠1 ≅ ∠3
2-8 Proving Angle Relationships
• Vertical Angles Theorem: If two angles are vertical
angles, then they are congruent
Right Angle Theorems:
• 2.9.1 ____________
Perpendicular lines intersect to form four right
angles
• 2.10 All right angles are __________
congruent
• 2.11 Perpendicular lines form congruent adjacent angles
• 2.12 If two angles are congruent and supplementary,
then each angle is a right angle
linear pair, then
• 2.13 If two congruent angles form a ______
they are right angles
Credits
• http://en.wikipedia.org/wiki/Euclid
• http://www.regentsprep.org/Regents/math/ge
ometry/GPB/theorems.htm
• http://www.regentsprep.org/Regents/math/ge
ometry/GPB/theorems.htm
• Google Images
• Geometry textbook
Jeopardy
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