Download Geometry - Eleanor Roosevelt High School

Proving Statements
in Geometry
Eleanor Roosevelt High School
Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Inductive Reasoning
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
?
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
?
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
?
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
?
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Visual Pattern
Describe and sketch the fourth figure in the
pattern:
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Mr. Chin-Sung Lin
ERHS Math Geometry
Number Pattern
Describe the pattern in the numbers and write
the next three numbers:
1
4
7
10
?
?
?
Mr. Chin-Sung Lin
ERHS Math Geometry
Number Pattern
Describe the pattern in the numbers and write
the next three numbers:
1
4
3
7
3
10 13 16
3
3
3
19
3
Mr. Chin-Sung Lin
ERHS Math Geometry
Number Pattern
Describe the pattern in the numbers and write
the next three numbers:
1
4
9
16
?
?
?
Mr. Chin-Sung Lin
ERHS Math Geometry
Number Pattern
Describe the pattern in the numbers and write
the next three numbers:
1
4
3
9
7
5
2
16 25 36
2
11
9
2
2
49
13
2
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjecture
An unproven statement that is based on observation
Mr. Chin-Sung Lin
ERHS Math Geometry
Inductive Reasoning
Inductive reasoning, or induction, is reasoning
from a specific case or cases and deriving a
general rule
You use inductive reasoning when you find a
pattern in specific cases and then write a
conjecture for the general case
Mr. Chin-Sung Lin
ERHS Math Geometry
Weakness of Inductive Reasoning
Direct measurement results can be only
approximate
We arrive at a generalization before we have
examined every possible example
When we conduct an experiment we do not
give explanations for why things are true
Mr. Chin-Sung Lin
ERHS Math Geometry
Strength of Inductive Reasoning
A powerful tool in discovering new
mathematical facts (making conjectures)
Inductive reasoning does not prove or explain
conjectures
Mr. Chin-Sung Lin
ERHS Math Geometry
Make a Conjecture
Give five colinear points, make a conjecture
about the number of ways to connect
different pairs of the points
Mr. Chin-Sung Lin
ERHS Math Geometry
Make a Conjecture
Give five colinear points, make a conjecture
about the number of ways to connect
different pairs of the points
No of Points
1
2
3
4
5
0
1
3
6
?
Picture
No of
Connections
Mr. Chin-Sung Lin
ERHS Math Geometry
Make a Conjecture
No of Points
1
2
3
4
5
0
1
3
6
?
Picture
No of
Connections
1
2
3
?
Mr. Chin-Sung Lin
ERHS Math Geometry
Make a Conjecture
No of Points
1
2
3
4
5
0
1
3
6
10
Picture
No of
Connections
1
2
3
4
Conjecture: You can connect five colinear
points 6 + 4 = 10 different ways
Mr. Chin-Sung Lin
ERHS Math Geometry
Prove a Conjecture
No of Points
1
2
3
4
5
0
1
3
6
10
Picture
No of
Connections
1
2
3
4
Conjecture: You can connect five colinear
points 6 + 4 = 10 different ways
Mr. Chin-Sung Lin
ERHS Math Geometry
Prove a Conjecture
To show that a conjecture is true, you must
show that it is true for all cases
Mr. Chin-Sung Lin
ERHS Math Geometry
Disprove a Conjecture
To show that a conjecture is false, you just
need to find one counterexample
A counterexample is a specific case for
which the conjecture is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise: Disprove a Conjecture
Conjecture: the sum of two number is
always greater than the larger number
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise: Disprove a Conjecture
Conjecture: the value of x2 always greater
than the value of x
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise: Disprove a Conjecture
Conjecture: the product of two numbers is
even, then the two numbers must both be
even
Mr. Chin-Sung Lin
ERHS Math Geometry
Analyzing Reasoning
Mr. Chin-Sung Lin
ERHS Math Geometry
Analyzing Reasoning
Use inductive reasoning to make conjectures
Use deductive reasoning to show that
conjectures are true or false
Mr. Chin-Sung Lin
ERHS Math Geometry
Analyzing Reasoning Example
What conclusion can you make about the product
of an even integer and any other integer
2 * 5 = 10
(-4) * (-7) = 28
2 * 6 = 12
6 * 15 = 90
use inductive reasoning to make a conjecture
Mr. Chin-Sung Lin
ERHS Math Geometry
Analyzing Reasoning Example
What conclusion can you make about the product
of an even integer and any other integer
2 * 5 = 10
(-4) * (-7) = 28
2 * 6 = 12
6 * 15 = 90
use inductive reasoning to make a conjecture
Conjecture:
Even integer * Any integer = Even integer
Mr. Chin-Sung Lin
ERHS Math Geometry
Analyzing Reasoning Example
Use deductive reasoning to show that a
conjecture is true
Conjecture:
Even integer * Any integer = Even integer
Let n and m be any integer
2n is an even integer since any integer multiplied by 2 is
even
(2n)m represents the product of an even interger and any
integer
(2n)m = 2(nm) is the product of 2 and an integer nm. So,
2nm is an even integer
Mr. Chin-Sung Lin
ERHS Math Geometry
Deductive Reasoning
Deductive reasoning, or deduction, is using
facts, definitions, accepted properties, and
the laws of logic to form a logical argument
While inductive reasoning is using specific
examples and patterns to form a conjecture
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions as
Biconditionals
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions as Biconditionals
• Right angles are angles with measure of 90
• Angles with measure of 90 are right angles
• When a conditional and its converse are both true:
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions as Biconditionals
• Right angles are angles with measure of 90
If angles are right angles, then their measure is 90
p  q (T)
• Angles with measure of 90 are right angles
• When a conditional and its converse are both true:
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions as Biconditionals
• Right angles are angles with measure of 90
If angles are right angles, then their measure is 90
p  q (T)
• Angles with measure of 90 are right angles
If measure of angles is 90, then their are right
angles
q  p (T)
• When a conditional and its converse are both true:
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions as Biconditionals
• Right angles are angles with measure of 90
If angles are right angles, then their measure is 90
p  q (T)
• Angles with measure of 90 are right angles
If measure of angles is 90, then their are right
angles
q  p (T)
• When a conditional and its converse are both true:
Angles are right angles if and only if their measure
is 90
q
p (T)
Mr. Chin-Sung Lin
ERHS Math Geometry
Deductive
Reasoning
Mr. Chin-Sung Lin
ERHS Math Geometry
Proofs
• A proof is a valid argument that establishes the
truth of a statement
• Proofs are based on a series of statements that are
assume to be true
• Definitions are true statements and are used in
geometric proofs
• Deductive reasoning uses the laws of logic to link
together true statements to arrive at a true
conclusion
Mr. Chin-Sung Lin
ERHS Math Geometry
Proofs of Euclidean Geometry
• given: The information known to be true
• prove: Statements and conclusion to be proved
• two-column proof:
• In the left column, we write statements that
we known to be true
• In the right column, we write the reasons why
each statement is true
* The laws of logic are used to deduce the conclusion but
the laws are not listed among the reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: In ΔABC, AB  BC
Prove: ΔABC is a right triangle
Proof:
Statements
1.AB  BC
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: In ΔABC, AB  BC
Prove: ΔABC is a right triangle
Proof:
Statements
Reasons
1.AB  BC
1. Given.
2.ABC is a right angle.
2. If two lines are perpendicular,
then they intersect to form right
angles.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: In ΔABC, AB  BC
Prove: ΔABC is a right triangle
Proof:
Statements
Reasons
1.AB  BC
1. Given.
2.ABC is a right angle.
2. If two lines are perpendicular,
then they intersect to form right
angles.
3.ΔABC is a right triangle.
3. If a triangle has a right angle then it is
a right triangle.
Mr. Chin-Sung Lin
ERHS Math Geometry
Paragraph Proof Example
Given: In ΔABC, AB  BC
Prove: ΔABC is a right triangle
Proof:
We are given that AB  BC. If two lines are perpendicular,
then they intersect to form right angles. Therefore, ABC is a
right angle. A right triangle is a triangle that has a right angle.
Since ABC is an angle of ΔABC, ΔABC is a right triangle.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements
1.BD is the bisector of ABC.
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements
1.BD is the bisector of ABC.
2.ABD
≅ DBC
Reasons
1. Given.
2. The bisector of an angle is a ray
whose endpoint is the vertex of the
angle and that divides the angle into
two congruent angles.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Example
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements
1.BD is the bisector of ABC.
2.ABD
≅ DBC
3.mABD = mDBC
Reasons
1. Given.
2. The bisector of an angle is a ray
whose endpoint is the vertex of the
angle and that divides the angle into
two congruent angles.
3. Congruent angles are angles that
have the same measure.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Exercise
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Exercise
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements
1.M is the midpoint of AMB.
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Exercise
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements
1.M is the midpoint of AMB.
2.AM
≅ MB
Reasons
1. Given.
2. The midpoint of a line segment is the
point of that line segment that
divides the segment into congruent
segments.
Mr. Chin-Sung Lin
ERHS Math Geometry
Two Column Proof Exercise
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements
1.M is the midpoint of AMB.
2.AM
≅ MB
3.AM = MB
Reasons
1. Given.
2. The midpoint of a line segment is the
point of that line segment that
divides the segment into congruent
segments.
3. Congruent segments are segments
that have the same measure.
Mr. Chin-Sung Lin
ERHS Math Geometry
Direct and Indirect
Proofs
Mr. Chin-Sung Lin
ERHS Math Geometry
Direct Proof
A proof that starts with the given statements and
uses the laws of logic to arrive at the statement
to be proved is called a direct proof
In most direct proofs we use definitions together
with the Law of Detachment to arrive at the
desired conclusion
All of the proofs we have learned so far are direct
proofs
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof
A proof that starts with the negation of the
statement to be proved and uses the laws of
logic to show that it is false is called an
indirect proof or a proof by contradiction
An indirect proof works because the negation of
the statement to be proved is false, then we
can conclude that the statement is true
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof
Let p be the given and q be the conclusion
1.
Assume that the negation of the conclusion (~q) is
true
2.
Use this assumption (~q is true) to arrive at a statement
that contradicts the given statement (p) or a true
statement derived from the given statement
3.
Since the assumption leads to a condiction, it (~q)must
be false. The negation of the assumption (q), the
desired conclusion, must be true
Mr. Chin-Sung Lin
ERHS Math Geometry
Direct Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Direct Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
1.mCDE ≠ 90
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Direct Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
1.mCDE ≠ 90
Reasons
1. Given.
2.CDE is not a right angle.
2. If the degree measure of an angle is
not 90, then it is not a right angle.
Mr. Chin-Sung Lin
ERHS Math Geometry
Direct Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
1.mCDE ≠ 90
Reasons
1. Given.
2.CDE is not a right angle.
2. If the degree measure of an angle is
not 90, then it is not a right angle.
3.CD is not perpendicular to
DE
3. If two intersecting lines do not form
right angles, then they are not
perpendicular.
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
1.CD is perpendicular to DE
Reasons
1. Assumption.
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
1.CD is perpendicular to DE
2.CDE is a right angle.
Reasons
1. Assumption.
2. If two intersecting lines are
perpendicular, then they form right
angles.
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
1.CD is perpendicular to DE
2.CDE is a right angle.
3.mCDE = 90.
Reasons
1. Assumption.
2. If two intersecting lines are
perpendicular, then they form right
angles.
3. If an angle is a right angle, then its
degree measure is 90
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
Reasons
1.CD is perpendicular to DE
2.CDE is a right angle.
3.mCDE = 90.
4.mCDE ≠ 90
1. Assumption.
2. If two intersecting lines are
perpendicular, then they form right
angles.
3. If an angle is a right angle, then its
degree measure is 90
4. Given
Mr. Chin-Sung Lin
ERHS Math Geometry
Indirect Proof Example
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements
Reasons
1.CD is perpendicular to DE
2.CDE is a right angle.
1. Assumption.
2. If two intersecting lines are
perpendicular, then they form right
angles.
3.mCDE = 90.
4.mCDE ≠ 90
3. If an angle is a right angle, then its
degree measure is 90
4. Given
5.CD is not perpendicular to
5. Contradiction in 3 and 4. Therefore, DE
the assumption is false and its
negation is true.
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates, Theorems,
and Proof
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulate (or Axiom)
A postulate (or axiom) is a statement whose truth
is accepted without proof
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem
A theorem is a statement that is proved by
deductive reasoning
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems and Geometry
applications
theorems
undefined
terms
defined
terms
postulates
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
Basic Properties of Equality
•
Reflexive Property
•
Symmetric Property
•
Transitive Property
Substitution Postulate
Partition Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
Addition Postulate
Subtraction Postulate
Multiplication Postulate
Division Postulate
Power Postulate
Roots Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Reflexive Property of Equality
A quantity is equal to itself
a=a
Algebraic example:
x=x
Mr. Chin-Sung Lin
ERHS Math Geometry
Reflexive Property of Equality
Geometric example:
The length of a segment is equal to itself
AB = AB
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Symmetric Property of Equality
An equality may be expressed in either order
If a = b, then b = a
Algebraic example:
x=5
then
5=x
Mr. Chin-Sung Lin
ERHS Math Geometry
Symmetric Property of Equality
Geometric example:
If the length of AB is equal to the length of CD, then
the length of CD is equal to the length of AB
AB = CD
then
CD = AB
A
B
C
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Transitive Property of Equality
Quantities equal to the same quantity are equal to
each other
If a = b and b = c, then a = c
Algebraic example:
x = y and y = 4
then
x=4
Mr. Chin-Sung Lin
ERHS Math Geometry
Transitive Property of Equality
Geometric example:
If the lengths of segments are equal to the length of
the same segment, they are equal to each other
AB = EF
then
AB = CD
and
EF = CD
A
B
E
F
C
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Substitution Postulate
A quantity may be substituted for its equal in any
statement of equality
Algebraic example:
x + y = 10 and y = 4x
then
x + 4x = 10
Mr. Chin-Sung Lin
ERHS Math Geometry
Substitution Postulate
Geometric example:
If the length of a segment is equal to the length of
another segment, it can be substituted by that
one in any statement of equality
AB = XY and
AB + BC = 10
then
XY + BC = 10
A
B
X
C
Y
Mr. Chin-Sung Lin
ERHS Math Geometry
Partition Postulate
A whole is equal to the sum of all its parts
•
•
A segment is congruent to the sum of its parts
An angle is congruent to the sum of its parts
Algebraic example:
2x + 3x = 5x
Mr. Chin-Sung Lin
ERHS Math Geometry
Partition Postulate
Geometric example:
The sum of all the parts of a segment is congruent
to the whole segment
AB + BC = AC
AB + BC = AC
A
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Addition Postulate
If equal quantities are added to equal quantities, the
sums are equal
•
•
If congruent segments are added to congruent
segments, the sums are congruent
If congruent angles are added to congruent angles,
the sums are congruent
If a = b and c = d, then a + c = b + d
Algebraic example:
x - 5 = 10
then x = 15
Mr. Chin-Sung Lin
ERHS Math Geometry
Addition Postulate
Geometric example:
If the length of a segment is added to two equal-length
segments, the sums are equal
AB ≅ CD and
BC ≅ BC
then
A
B
C
D
AB + BC ≅ CD + BC
Mr. Chin-Sung Lin
ERHS Math Geometry
Subtraction Postulate
If equal quantities are subtracted from equal quantities,
the differences are equal
•
•
If congruent segments are subtracted to congruent
segments, the differences are congruent
If congruent angles are subtracted to congruent
angles, the differences are congruent
If a = b and c = d, then a - c = b - d
Algebraic example:
x + 5 = 10
then x = 5
Mr. Chin-Sung Lin
ERHS Math Geometry
Subtraction Postulate
Geometric example:
If a segment is subtracted from two congruent
segments, the differences are congruent
AC ≅ BD and
BC ≅ BC
A
then
AC - BC ≅ BD - BC
B
C
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Multiplication Postulate
If equal quantities are multiplied by equal
quantities, the products are equal
•
Doubles of equal quantities are equal
If a = b, and c = d, then ac = bd
Algebraic example:
x = 10
then
2x = 20
Mr. Chin-Sung Lin
ERHS Math Geometry
Multiplication Postulate
Geometric example:
If the lengths of two segments are equal, their like
multiples are equal
AO = CP
then
2AO = 2CP
A
O
B
C
P
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Division Postulate
If equal quantities are divided by equal nonzero
quantities, the quotients are equal
•
Halves of equal quantities are equal
If a = b, and c = d, then a / c = b / d (c ≠ 0 and d ≠ 0)
Algebraic example:
2x = 10
then
x=5
Mr. Chin-Sung Lin
ERHS Math Geometry
Division Postulate
Geometric example:
If the lengths of two segments are congruent, their
like divisions are congruent
AB = CD
then
½ AB = ½ CD
A
O
B
C
P
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Powers Postulate
The squares of equal quantities are equal
If a = b, and a2 = b2
Algebraic example:
x = 10
then
x2 = 100
Mr. Chin-Sung Lin
ERHS Math Geometry
Powers Postulate
Geometric example:
If the lengths of two hypotenuses are equal, their
powers are equal
AB = XY
then
AB2 = XY2
A
C
X
B Z
Y
Mr. Chin-Sung Lin
ERHS Math Geometry
Root Postulate
Positive square roots of positive equal quantities are
equal
If a = b, and a > 0, then √a = √b
Algebraic example:
x = 100
then
√x = 10
Mr. Chin-Sung Lin
ERHS Math Geometry
Root Postulate
Geometric example:
If the squares of the lengths of two hypotenuses
are equal, their square roots are equal
AB2 = XY2
then
AB = XY
A
C
X
B Z
Y
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Postulates
for Proofs
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
Which postulate tells us that if the measures of
two angles are equal to a third angle’s, then
they are equal to each other?
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
Which postulate tells us that if the measures of
two angles are equal to a third angle’s, then
they are equal to each other?
Transitive Property
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
Which postulate tells us that the measure of an
angle is equal to itself?
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
Which postulate tells us that the measure of an
angle is equal to itself?
Reflexive Property
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If BD is equal to AE, and AE is equal to EC, how
do we know that BD is equal to EC?
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If BD is equal to AE, and AE is equal to EC, how
do we know that BD is equal to EC?
Transitive Property
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
How do we know that BAF + FAC is equal to
BAC?
B
F
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
How do we know that BAF + FAC is equal to
BAC?
B
F
A
C
Partition Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If BA is equal to FA, and EC is equal to AD, how
do we know BA / EC = FA / AD?
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If BA is equal to FA, and EC is equal to AD, how
do we know BA / EC = FA / AD?
Division Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If AF is equal to AC, how do we know that AF - BD
= AC - BD?
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If AF is equal to AC, how do we know that AF - BD
= AC - BD?
Subtraction Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If segment AB is congruent to segment XY and
segment BC is congruent to segment YZ, how
do we know that AB + BC = XY + YZ
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates for Proofs
If segment AB is congruent to segment XY and
segment BC is congruent to segment YZ, how
do we know that AB + BC = XY + YZ
Addition Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates
for Proofs
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
If AB  BC and LM  MN, prove mABC = mLMN
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
If AB  BC and LM  MN, prove mABC = mLMN
Given: AB  BC and LM  MN
Prove: mABC = mLMN
L
A
B
C
M
N
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB  BC and LM  MN
Prove: mABC = mLMN
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB  BC and LM  MN
Prove: mABC = mLMN
Proof:
Statements
1.AB  BC and LM  MN
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB  BC and LM  MN
Prove: mABC = mLMN
Proof:
Statements
Reasons
1.AB  BC and LM  MN
1. Given.
2.ABC and LMN are right
angles.
2. Perpendicular lines are two
lines intersecting to form right
angles.
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB  BC and LM  MN
Prove: mABC = mLMN
Proof:
Statements
Reasons
1.AB  BC and LM  MN
1. Given.
2.ABC and LMN are right
angles.
2. Perpendicular lines are two
lines intersecting to form right
angles.
3.
3. A right angle is an angle whose
degree measure is 90
mABC = 90, mLMN = 90
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB  BC and LM  MN
Prove: mABC = mLMN
Proof:
Statements
Reasons
1.AB  BC and LM  MN
1. Given.
2.ABC and LMN are right
angles.
2. Perpendicular lines are two
lines intersecting to form right
angles.
3.
3. A right angle is an angle whose
degree measure is 90
mABC = 90, mLMN = 90
4.90 = mLMN
4. Symmetric property of equality
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB  BC and LM  MN
Prove: mABC = mLMN
Proof:
Statements
Reasons
1.AB  BC and LM  MN
1. Given.
2.ABC and LMN are right
angles.
2. Perpendicular lines are two
lines intersecting to form right
angles.
3.
mABC = 90, mLMN = 90
3. A right angle is an angle whose
degree measure is 90
4.
90 = mLMN
4. Symmetric property of equality
5.
mABC = mLMN
5. Transitive property of equality
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
If AB = 2 CD, and CD = XY, prove AB = 2 XY
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
If AB = 2 CD, and CD = XY, prove AB = 2 XY
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
A
B C
D
X
Y
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
Proof:
Statements
1.AB = 2 CD, and CD = XY
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
Proof:
Statements
Reasons
1.AB = 2 CD, and CD = XY
1. Given.
2.AB = 2 XY
2. Substitution postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
If ABCD are collinear, AB = CD, prove AC = BD
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
If ABCD are collinear, AB = CD, prove AC = BD
Given: ABCD and AB = CD
Prove: AC = BD
A
B
C
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements
1.AB = CD
Reasons
1. Given.
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements
Reasons
1.AB = CD
1. Given.
2.BC = BC
2. Reflexive property of equality
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements
Reasons
1.AB = CD
1. Given.
2.BC = BC
2. Reflexive property of equality
3.AB + BC = CD + BC
3. Addition postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements
Reasons
1.AB = CD
1. Given.
2.BC = BC
2. Reflexive property of equality
3.AB + BC = CD + BC
4.AB + BC = AC
3. Addition postulate
4. Partition postulate
CD + BC = BD
Mr. Chin-Sung Lin
ERHS Math Geometry
Apply Postulates for Proofs
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements
Reasons
1.AB = CD
1. Given.
2.BC = BC
2. Reflexive property of equality
3.AB + BC = CD + BC
4.AB + BC = AC
3. Addition postulate
4. Partition postulate
CD + BC = BD
5.
AC = BD
5. Substitution postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Q&A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin