Transcript Geometry Chapter 2 Lessons

Geometry
Chapter 2
Conditional Statements
A statement is a sentence whose truth value can be
determined
A conditional statement is a type of logical statement in the form of if → then
If a bird is a pelican, then it eats fish.
p
hypothesis
→
q
conclusion
The conclusion is assured to happen only on
the condition the hypothesis has been met.
Converse of conditional statement
The converse of a conditional
statement has the hypothesis and
the conclusion reversed.
Truth value of a converse
statement is not necessarily the
same as the original statement.
If a bird eats fish then it is a pelican.
q
hypothesis
→
p
conclusion
What does true or false mean?
True only if all cases true.
False if one counter example exists.
Biconditional Statements
A biconditional statement is one in
which the original conditional
statement and its converse have the
same truth value.
If the sun is in the west, then it is afternoon.
If it is afternoon, then the sun is in the west.
combined statement
task, write the converse of this statement
The sun is in the west if and only if it is afternoon.
iff
The sun is in the west iff it is afternoon.
p
↔
q
All apples are a fruit.
practice, write the following as a conditional statement
If a food is an apple then it is a fruit.
Truth Value?
practice, write the converse of the conditional statement
If a food is a fruit then it is an apple.
Truth Value?
Is this a biconditional?
Assignment 81/1 – 4, 9 – 20
Basic Postulates
P5 – Through any two distinct points, there exists exactly one line.
P6 – A line contains at least two points
Basic Postulates
P7 – Through any three non collinear points, there exists exactly
one plane.
P8 – A plane contains at least 3 non-collinear points
Basic Postulates
P9 – If two distinct points lie in a plane, then the line containing
them lies in the plane.
Basic Postulates
P10 – If two distinct planes intersect, their intersection is a line.
80/communicate about
geometry A – F
82/21 – 24, 27 – 33
83/mixed review 1 – 16
Reasoning with Properties of Algebra
•
•
•
•
•
•
•
•
•
Addition Property
Subtraction Property
Multiplication Property
Division Property
Distributive Property
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
Addition Property of Equality
If a = b, then a + c = b + c
Adding the same value to equivalent expressions maintains the equality
If NK = 8, then NK + 4 = 12
or
statement
Each side is increased by 4.
reason
1. NK = 8
2. NK + 4 = 12
addition property of equality
Subtraction Property of Equality
If a = b, then a - c = b - c
Subtracting the same value from equivalent expressions maintains the equality
If mA = 55, then mA - 5 = 50
or
statement
Each side is decreased by 5.
reason
1. mA = 55
2. mA - 5 = 50
subtraction property of equality
Multiplication Property of Equality If a = b, then ac = bc
Multiplying equivalent expressions by the same value maintains the equality
If mA = 36, then 2(mA) = 72
or
statement
Each side multiplied by 2
reason
1. mA = 36
2. 2(mA) = 72
multiplication property of equality
If a = b, then a ÷ c = b ÷ c
Division Property of Equality
Dividing equivalent expressions by the same value to maintains the equality
If YD = 15, then YD/3 = 5
or
statement
Each side divided by 3
reason
1. YD = 15
2. YD/3 = 5
division property of equality
Distributive property of
multiplication over addition
ab + ac = a(b + c)
The distributive property in reverse allows us to combine like terms
statement
reason
1. 3x + 6x
2. x(3 + 6)
distributive property
3. x(9)
number fact
4. 9x
symmetric propertry
Reflexive Property of Equality
for any real number a = a
Any number is equal to itself
A
B
statement
1. AB = AB
reason
reflexive property of equality
Symmetric Property of Equality
If a = b, then b = a
Equivalent expressions maintains their equality regardless of order
Sally likes Vijay and Vijay likes Sally
A
B
C
D
If AB = CD then CD = AB
or
statement
reason
1. AB = CD
2. CD = AB
symmetric property of equality
Transitive Property of Equality
If a = b, and b = c, then a = c
Two expressions equal to the same expression are equal to each other.
A
B
C
If mA = mB and mB = mC then mA = mC
A and C are both equivalent to B so they are equal to each other
or
statement
reason
1. mA = mB
2. mB = mC
3. mA = mC
transitive property of equality
Substution Property of Equality
If a = b, and a ± c = d, then b ± c = d
If two expressions are equivalent, one can replace the other in any equation
If AB = CD, and AB + BC = AC, then CD + BC = AC
or
statement
CD replaces AB
reason
1. AB = CD
2. AB + BC = AC
segment addition postulate
3. CD + BC = AC
substitution property of equality
2 Column Proof Format
Given: AC = BD
Prove: AB = CD
A
B
statement
C
D
reason
1.1. AC
Given
= BD
statement
•
1.
Given
1. Given
2. AB + BC = AC
•
2.
Segment Addition Postulate (from picture)
Definitions
3. BC + CD = BD
•
3.
Segment Addition Postulate
Postulates
4. AB + BC = BD
•
4.
Substitution
Property
Algebraic
Properties
5. AB + BC = BC + CD
•
5.
Transitive Property
Theorems
6.
AB = CD
#. Prove
statement
•
6.
Property
#. Subtraction
reason varies
Structure of a Logical Argument
1. Theorem – Hypothesis, Conclusion
2. Argument body – Series of logical
statements, beginning with the Hypothesis and
ending with the Conclusion.
3. Restatement of the Theorem. (I told you so)
If you are careless with fire, Then a fish will die.
•
•
•
•
•
•
•
•
If you are careless with fire,
Then there will be a forest fire.
If there is a forest fire,
Then there will be nothing to trap the rain.
If there is nothing to trap the rain,
Then the mud will run into the river.
If the mud runs into the river,
Then the gills of the fish will get clogged with silt
If the gills of the fish get clogged with silt,
Then the fish can’t breathe.
If a fish can’t breath,
Then a fish will die
If you are careless with fire, Then a fish will die.
2 Column Proof Format
Given: AC = BD
Theorem
Prove: AB = CD
B
A
statement
C
D
Argument Body
reason
1. AC = BD
•
1. Given
2. AB + BC = AC
•
2. Segment Addition Postulate (from picture)
3. BC + CD = BD
•
3. Segment Addition Postulate
4. AB + BC = BD
•
4. Substitution Property
5. AB + BC = BC + CD •
5. Transitive Property
6. AB = BD
6. Subtraction Property
•
Restating Theorem left out
Angle Relationships
•
•
•
•
•
•
•
•
Vertical Angles
Linear Pair (of angles)
Complementary Angles
Supplementary Angles
Linear Pair Postulate
Congruent Supplements Theorem
Congruent Complements Theorem
Vertical Angles Theorem
Vertical Angles
Vertical Angles are the non adjacent angles formed by two intersecting lines.
1 & 2 are a pair of vertical angles.
3
2
1
4
3 & 4 are also a pair of vertical angles.
6
5
5 & 6 are not a pair of vertical angles.
Linear Pair
If the noncommon sides of adjacent angles are opposite rays then the angles
are a linear pair.
1
statement
2
reason
from picture
4. 1 & 2 are a
linear pair.
4. Definition of linear
pair.
Complementary Angles
If the sum of the measures of two angles is 90, then the angles are complementary.
Each angle is the complement of the other
1
2
If m1 + m2 = 90, then the angles are complementary
statement
reason
1. m1 + m2 = 90
Reversible
2. 1 & 2 are
complementary
2. Definition of Complementary
angles.
Supplementary Angles
If the sum of the measures of two angles is 180, then the angles are supplementary.
1
2
Each angle is the supplement of the other
If m1 + m2 = 180, then the angles are supplementary
statement
Reversible
reason
1. m1 + m2 = 180
2. 1 & 2 are
supplementary
2. Definition of Supplementary
angles.
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary. (m1 + m2 = 180)
1
statement
•
•
•
from picture
4. 1 & 2 are a linear
pair.
5. m1 + m2 = 180
2
reason
• 4. Definition of linear pair.
• 5. Linear Pair Postulate
1. Solve:
x  4 3x  2

12
28
2. If the product of the slopes of two lines is -1, then the lines
are perpendicular.
a) write the converse
b) write the statement represented by p↔q
3. Write an example of the transitive property.
Congruent Supplements Theorem
If two angles are supplementary to the same or to congruent angles, then they are
congruent.
A
B
C
If A & C are supplementary and B & C are supplementary
then A & B are congruent.
statement
•
•
•
1. A & C are supplementary
2. B & C are supplementary
3. A  B
reason
3. Congruent Supplements Theorem
Congruent Supplements Theorem
If two angles are supplementary to the same or to congruent angles, then they are
congruent.
A
B
C
D
If A & C are supplementary and B & D are supplementary
and C & D are congruent.
statement
•
•
•
•
1. A & C are supplementary
2. B & D are supplementary
3. C  D
4 A  B
then A & B are congruent.
reason
4. Congruent Supplements Theorem
Congruent Supplements Theorem
If two angles are supplementary to the same or to congruent angles, then they are
congruent.
A
B
C
Given: A & C are supplementary
B & C are supplementary
Prove: A   B
statement
•
•
•
•
•
1. A & C also, B & C are
supplementary
2. mA + mC = 180
mB + mC = 180
3. mA + mC = mB + mC
4. mA = mB
5. A   B
reason
•
1. Given
•
2. Def. of Supplementary
angles
3. Transitive Prop. of =
4. Subtraction Prop of =
5. Def. of Congruent angles
•
•
•
Congruent Complements Theorem
If two angles are complementary to the same or to congruent angles, then they are
congruent.
A
B
C
D
If A & C are complementary and B & D are complementary
and C & D are congruent.
statement
•
•
•
•
1. A & C are complementary
2. B & D are complementary
3. C  D
4 A  B
then A & B are congruent.
reason
4. Congruent Complements Theorem
Congruent Complements Theorem
If two angles are complementary to the same or to congruent angles, then they are
congruent.
A
B
C
If A & C are complementary and B & C are complementary
then A & B are congruent.
statement
•
•
•
1. A & C are complementary
2. B & C are complementary
3. A  B
reason
3. Congruent Complements Theorem
Vertical Angles Theorem
If two angles are vertical, then they are congruent.
m
3
Given: lines m  l at point A
Prove: 1  2
statement
•
•
•
•
•
•
1. lines intersect at A
2. 1 & 3 are a linear pair
3 & 2 are a linear pair
3. m1 + m3 = 180
m3 + m2 = 180
4. m1 + m3 = m3 + m2
5. m1 = m2
6. 1  2
2
1
A
l
reason
•
•
1. Given
2. Def. of Linear Pair (picture)
•
3. Linear Pair Postulate
•
•
•
4. Transitive Property of =
5. Subtraction Prop. of =
6. Def. of congruent angles
Vertical Angles and Linear Pair Postulate Applications
What type of angles?
10x + 40
What is the relationship?
20x - 50
What is the equation?
m1 = m2
10x + 40 = 20x - 50
90 = 10x
x=9
m = 10(9) + 40 = 130
Vertical Angles and Linear Pair Postulate Applications
What type of angles?
10x + 40
20x - 50
What is the relationship?
What is the equation?
m1 + m2 = 180
10x + 40 + 20x – 50 = 180
30x -10 = 180
30x = 190
x = 19/3
m = 10(19/3) + 40 = 310/3 = 103 1/3
m1 = 2x + 15
m2 = 6x – 5
1
m1 = 2x + 15 = 2(10) + 15 = 35
2
m2 = 6x – 5 = 6(10) – 5 = 55
1 & 2 are complementary
m 1 + m 2 = 90
(2x + 15) + (6x – 5) = 90
8x + 10 = 90
8x = 80
X = 10