Transcript 1 2 - spartansmath

Geometry 1
Unit 2: Reasoning and Proof
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Geometry 1 Unit 2
2.1 Conditional Statements
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Conditional Statements

Conditional StatementA

statement with two parts
If-then form
A
way of writing a conditional statement that clearly
showcases the hypothesis and conclusion

Hypothesis The

“if” part of a conditional Statement
Conclusion
 The
“then” part of a conditional Statement
3
Conditional Statements

Examples of Conditional Statements
 If
today is Saturday, then tomorrow is Sunday.
 If it’s a triangle, then it has a right angle.
 If x2 = 4, then x = 2.
 If you clean your room, then you can go to the
mall.
 If p, then q.
4
Conditional Statements


Example 1
Circle the hypothesis and underline the conclusion in
each conditional statement

If you are in Geometry 1, then you will learn about the building
blocks of geometry

If two points lie on the same line, then they are collinear

If a figure is a plane, then it is defined by 3 distinct points
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Conditional Statements


Example 2
Rewrite each statement in if…then form
A
line contains at least two points
If a figure is a line, then it contains at least two points
 When
two planes intersect their intersection is a line
If two planes intersect, then their intersection is a
line.
 Two
angles that add to 90° are complementary
If two angles add to equal 90°, then they are
complementary.
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Conditional Statements

Counterexample
 An
example that proves that a given
statement is false

Write a counterexample
 If x2
= 9, then x = 3
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Conditional Statements

Example 3
 Determine
if the following statements are true
or false.
 If false, give a counterexample.
If x + 1 = 0, then x = -1
 If a polygon has six sides, then it is a decagon.
 If the angles are a linear pair, then the sum of the
measure of the angles is 90º.

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Conditional Statements

Negation
 In
most cases you can form the negation of a
statement by either adding or deleting the
word “not”.
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Conditional Statements

Examples of Negations
 Statement:
mA  30
 Negation
:
mA  30
 Statement:
Mr. Ross is not more than 6 feet tall.
 Negation: Mr. Ross is more than 6 feet tall
 I am doing my homework.
 Negation:
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Conditional Statements
Example 4
 Write the negation of each statement.
Determine whether your new statement is
true or false.


Stanfield is the largest city in Arizona.
 All triangles have three sides.
 Dairy cows are not purple.
 Some VGHS students have brown hair.
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Conditional Statements

Converse

Formed by switching the if and the then part.

Original


If you like green, then you will love my new shirt.
Converse

If you love my new shirt, then you like green.
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Conditional Statements

Inverse
 Formed
by negating both the if and the then
part.

Original


If you like green, then you will love my new shirt.
Inverse

If you do not like green, then you will not love my new
shirt.
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Conditional Statements

Contrapositive
 Formed
by switching and negating both the if
and then part.

Original


If you like green, then you will love my new shirt.
Contrapositive

If you do not love my new shirt, then you do not like
green.
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Conditional Statements
Write in if…then form.
 Write the converse, inverse and
contrapositive of each statement.

I
will wash the dishes, if you dry them.
A
square with side length 2 cm has an area of
4 cm2.
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Conditional Statements

Point-line postulate:
 Through
any two points, there exists exactly
one line

Point-line converse:
A

line contains at least two points
Intersecting lines postulate:
 If
two lines intersect, then their intersection is
exactly one point
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Conditional Statements

Point-plane postulate:
 Through
any three noncollinear points there exists
one plane

Point-plane converse:
A

plane contains at least three noncollinear points
Line-plane postulate:
 If
two points lie in a plane, then the line containing
them lies in the plane

Intersecting planes postulate:
 If
two planes intersect, then their intersection is a line
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Geometry 1 Unit 2
2.2: Definitions and Biconditional
Statements
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Definitions and
Biconditional Statements
Can be rewritten with “If and only if”
 Only occurs when the statement and the
converse of the statement are both true.
 A biconditional can be split into a
conditional and its converse.

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Definitions and
Biconditional Statements

Example 1
 An
A
angle is right if and only if its measure is 90º
number is even if and only if it is divisible by two.
A
point on a segment is the midpoint of the segment if
and only if it bisects the segment.
 You
attend school if and only if it is a weekday.
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Definitions and
Biconditional Statements

Perpendicular lines
 Two
lines are perpendicular if they intersect to
form a right angle

A line perpendicular to a plane
A
line that intersects the plane in a point and is
perpendicular to every line in the plane that
intersects it

The symbol
to.

is read, “is perpendicular
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Definitions and
Biconditional Statements

Example 2
 Write
the definition of perpendicular as
biconditional statement.
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Definitions and
Biconditional Statements

Example 3
 Give
a counterexample that demonstrates
that the converse is false.

If two lines are perpendicular, then they intersect.
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Definitions and
Biconditional Statements

Example 4
 The
following statement is true. Write the
converse and decide if it is true or false. If the
converse is true, combine it with its original to
form a biconditional.

If x2 = 4, then x = 2 or x = -2
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Definitions and
Biconditional Statements

Example 5
 Consider
the statement
x2 < 49 if and only if x < 7.
Is this a biconditional?
 Is the statement true?

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Geometry 1 Unit 2
2.3 Deductive Reasoning
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Deductive Reasoning

Symbolic Logic
 Statements
are replaced with variables, such
as p, q, r.
 Symbols are used to connect the statements.
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Deductive Reasoning
Symbol
~
Λ
V
→
↔
Meaning
not
and
or
if…then
if and only if
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Deductive Reasoning

Example 1
 Let
p be “the sum of the measure of two
angles is 180º” and
 Let q be “two angles are supplementary”.

What does p → q mean?

What does q → p mean?
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Deductive Reasoning

Example 2
 p:
Jen cleaned her room.
 q: Jen is going to the mall.
What does p → q mean?
 What does q → p mean?
 What does ~q mean?
 What does p Λ q mean?

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Deductive Reasoning

Example 3
 Given
t and s, determine the meaning of the
statements below.
 t: Jeff has a math test today
 s: Jeff studied
tVs
 s → t
 ~s → t

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Deductive Reasoning

Deductive Reasoning
 Deductive
reasoning uses facts, definitions,
and accepted properties in a logical order to
write a logical argument.
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Deductive Reasoning

Law of Detachment
 When
you have a true conditional statement
and you know the hypothesis is true, you can
conclude the conclusion is true.
Given:
Given:
Conclusion:
p→q
p
q
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Deductive Reasoning

Example 4
 Determine
if the argument is valid.
If Jasmyn studies then she will ace her test.
Jasmyn studied.
Jasmyn aced her test.
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Deductive Reasoning

Example 5
 Determine
if the argument is valid.
If Mike goes to work, then he will get home late.
Mike got home late.
Mike went to work
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Deductive Reasoning

Law of Syllogism
 Given
two linked conditional statements you
can form one conditional statement.
Given:
Given:
Conclusion:
p→q
q→r
p→r
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Deductive Reasoning

Example 6
 Determine
if the argument is valid.
If today is your birthday, then Joe will bake a cake.
If Joe bakes a cake, then everyone will celebrate.
If today is your birthday, then everyone will
celebrate.
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Deductive Reasoning

Example 7
 Determine
if the argument is valid.
If it is a square, then it has four sides.
If it has four sides, then it is a quadrilateral.
If it is a square, then it is a quadrilateral.
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Geometry 1 Unit 2
2.4 Reasoning with Properties
from Algebra
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Reasoning with
Properties from Algebra

Objectives
 Review
of algebraic properties
 Reasoning
 Applications
of properties in real life
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Reasoning with
Properties from Algebra

Addition property
 If

Subtraction property
 If

a = b, then a – c = b – c
Multiplication property
 If

a = b, then a + c = b + c
a = b, then ac = bc
Division property
 If
a = b, then
a c  b c
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Reasoning with
Properties from Algebra

Reflexive property
 For

Symmetric property
 If

a=b, then b = a
Transitive Property
 If

any real number a, a = a
a = b and b = c, then a = c
Substitution property
 If
a = b, then a can be substituted for b in any
equation or expression

Distributive property
 2(x
+ y) = 2x + 2y
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Reasoning with
Properties from Algebra

Example 1
 Solve
6x – 5 = 2x + 3 and write a reason for each step
Statement
6x – 5 = 2x + 3
Reason
Given
4x – 5 = 3
4x = 8
x=2
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Reasoning with
Properties from Algebra
Example 2
 2(x – 3) = 6x + 6
1.

2.

3.

4.

5.
Given
44
Reasoning with
Properties from Algebra

Determine if the equations are valid or invalid.
 (x
+ 2)(x + 2) = x2 + 4
 x3x3
 -(x
= x6
+ y) = x – y
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Reasoning with
Properties from Algebra

Geometric Properties of Equality
 Reflexive

property of equality
For any segment AB, AB = AB
 Symmetric
property of equality
 If mA  mB, then mB  mA
 Transitive property of equality

If AB = CD and CD = EF, then, AB = EF
46
Reasoning with
Properties from Algebra
Example 3
A B
C
D
In the diagram, AB = CD. Show that AC = BD
Statement
Reason
AB = CD
AB + BC = BC + CD
AC = AB + BC
BD = BC + CD
AC = BD
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A
In the diagram, ABC =
DBF. Show that
 ABD = CBF
C
D
B
F
Statement
Reason
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Geometry 1 Unit 2
2.5: Proving Statements about
Segments
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Proving Statements about
Segments

Key Terms:
 2-column

proof
A way of proving a statement using a numbered
column of statements and a numbered column of
reasons for the statements
 Theorem

A true statement that is proven by other true
statements
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Proving Statements about
Segments

Properties of Segment Congruence
 Reflexive

For any segment AB, AB  AB
 Symmetric

If AB  CD, then CD  AB
 Transitive

If AB  EF and AB  CD ,then CD  EF
51
Proving Statements about
Segments

Example 1
K
 In
triangle JKL,
Given: LK = 5, JK = 5, JK = JL
Prove: LK = JL
J
L
Statement
1.
Reason
1. Given
2.
3.
4.
5.
2. Given
3. Transitive property of equality
4.
5. Given
6.
6. Transitive property of congruence
52
Proving Statements about
Segments


Duplicating a Segment
Tools


Straight edge: Ruler or piece
of wood or metal used for
creating straight lines
 Compass: Tool used to create
arcs and circles
Steps
1.
2.
3.
A
C
B
D
4.
5.
6.
Use a straight edge to
draw a segment longer
than segment AB
Label point C on new
segment
Set compass at length of
segment AB
Place compass point at C
and strike an arc on line
segment
Label intersection of arc
and segment point D
Segment CD is now
congruent to segment AB53
Geometry 1 Unit 2
2.6: Proving Statements about
Angles
54
Proving Statements about Angles

Properties of Angle Congruence
 Reflexive
For any angle A, A  A.
 Symmetric
If A  B, thenB  A.
 Transitive
If A  Band B  C , thenA  C.
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Proving Statements about Angles

Right Angle Congruence Theorem
 All
right angles are congruent.
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Proving Statements about Angles

Congruent Supplements Theorem
 If
two angles are supplementary to the same angle,
then they are congruent.
If
m1  m2  180 and
m2  m3  180 ,
then1  3.
1
2
3
57
Proving Statements about Angles

Congruent Complements Theorem
 If
two angles are complementary to the same angle,
then the two angles are congruent.
If
m4  m5  90 and
m5  m6  90 ,
then4  6.
5
6
4
58
Proving Statements about Angles

Linear Pair Postulate
 If
two angles form a linear pair, then they are
supplementary.
m1  m2  180
1
2
59
Proving Statements about Angles

Vertical Angles Theorem
 Vertical
angles are congruent
2
1
3
4
1  3, 2  4
60
Proving Statements about Angles

Example 1
 Given:1  2, 3  4, 2  3.
B
 Prove:1  4
1
A
Statement
1.
2.
3.
4.
Reason
1.
2.
3.
4.
3
2
4
C
61
Proving Statements about Angles

Example 2
 Given: m1  63 , 1  3, 3  4
 Prove:
m4  63
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
1
3
2
4
62
Proving Statements about Angles

Example 3
 Given: DAB, ABC are right angles
D
ABC  BDC
 Prove:
DAB  BDC
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
A
C
B
63
Proving Statements about Angles

Example 4


Given:
m1 = 24º,
m3 = 24º
1 and 2 are
complementary
3 and 4 are
complementary
Prove: 2  4
1 2
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
3
4
64
Proving Statements about Angles

Example 5
the diagram m1 = 60º and BFD is right.
Explain how to show m4 = 30º.
 In
C
B
D
1
A
2 3
F
4
E
65
Proving Statements about Angles

Example 6
1 and 2 are
a linear pair, 2 and
3 are a linear pair
 Prove: 1  3
 Given:
1
2
3
Statement
Reason
1.
1.
2.
2.
3.
3.
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