proving statements in geometry
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Proving Statements in Geometry
Eleanor Roosevelt High School
Geometry
Mr. Chin-Sung Lin
Inductive Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1
?Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1 Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1
?Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1 Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1
?Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1 Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1
?Fig. 2 Fig. 3 Fig. 4
Visual Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe and sketch the fourth figure in the pattern:
Fig. 1 Fig. 2 Fig. 3 Fig. 4
Number Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe the pattern in the numbers and write the next three numbers:
1 ?4 7 10 ? ?
Number Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe the pattern in the numbers and write the next three numbers:
1 4 7 10 13 16 19
3 3 3 3 3 3
Number Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe the pattern in the numbers and write the next three numbers:
1 ?4 9 16 ? ?
Number Pattern
ERHS Math Geometry
Mr. Chin-Sung Lin
Describe the pattern in the numbers and write the next three numbers:
25 36 49
3 5 7 9 11 13
1 4 9 16
2 2 2 2 2
Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
An unproven statement that is based on observation
Inductive Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Weakness of Inductive Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Strength of Inductive Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
A powerful tool in discovering new mathematical facts (making conjectures)
Inductive reasoning does not prove or explain conjectures
Make a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points
Make a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Picture
No of Connections 0 1 3 6 ?
Make a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
No of Points 1 2 3 4 5
Picture
No of Connections 0 1 3 6 ?
1 2 3 ?
Make a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
No of Points 1 2 3 4 5
Picture
No of Connections 0 1 3 6 10
Conjecture: You can connect five colinear points 6 + 4 = 10 different ways
1 2 3 4
Prove a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
No of Points 1 2 3 4 5
Picture
No of Connections 0 1 3 6 10
Conjecture: You can connect five colinear points 6 + 4 = 10 different ways
1 2 3 4
Prove a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
To show that a conjecture is true, you must show that it is true for all cases
Disprove a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Exercise: Disprove a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
Conjecture: the sum of two number is always greater than the larger number
Exercise: Disprove a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
Conjecture: the value of x2 always greater than the value of x
Exercise: Disprove a Conjecture
ERHS Math Geometry
Mr. Chin-Sung Lin
Conjecture: the product of two numbers is even, then the two numbers must both be even
Analyzing Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
Analyzing Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
Use inductive reasoning to make conjectures
Use deductive reasoning to show that conjectures are true or false
Analyzing Reasoning Example
ERHS Math Geometry
Mr. Chin-Sung Lin
What conclusion can you make about the product of an even integer and any other integer2 * 5 = 10 (-4) * (-7) = 282 * 6 = 12 6 * 15 = 90
use inductive reasoning to make a conjecture
Analyzing Reasoning Example
ERHS Math Geometry
Mr. Chin-Sung Lin
What conclusion can you make about the product of an even integer and any other integer2 * 5 = 10 (-4) * (-7) = 282 * 6 = 12 6 * 15 = 90
use inductive reasoning to make a conjecture
Conjecture: Even integer * Any integer = Even integer
Analyzing Reasoning Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Use deductive reasoning to show that a conjecture is true
Conjecture: Even integer * Any integer = Even integer
Let n and m be any integer2n 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
Deductive Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Definitions as Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
Definitions as Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
• 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:
Definitions as Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
• Right angles are angles with measure of 90If angles are right angles, then their measure is 90p q (T)
• Angles with measure of 90 are right angles
• When a conditional and its converse are both true:
Definitions as Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
• Right angles are angles with measure of 90If angles are right angles, then their measure is 90p q (T)
• Angles with measure of 90 are right angles If measure of angles is 90, then their are right anglesq p (T)
• When a conditional and its converse are both true:
Definitions as Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
• Right angles are angles with measure of 90If angles are right angles, then their measure is 90p q (T)
• Angles with measure of 90 are right angles If measure of angles is 90, then their are right anglesq p (T)
• When a conditional and its converse are both true:Angles are right angles if and only if their measure is 90q p (T)
Deductive Reasoning
ERHS Math Geometry
Mr. Chin-Sung Lin
Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
• 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
Proofs of Euclidean Geometry
ERHS Math Geometry
Mr. Chin-Sung Lin
• 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
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: In ΔABC, AB BC
Prove: ΔABC is a right triangle
Proof:
Statements Reasons
1.AB BC 1. Given.
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
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.
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
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.
Paragraph Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
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.
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements Reasons
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements Reasons
1.BD is the bisector of ABC. 1. Given.
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements Reasons
•BD is the bisector of ABC. 1. Given.
ABD ≅ DBC 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.
Two Column Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: BD is the bisector of ABC.
Prove: mABD = mDBC
Proof:
Statements Reasons
•BD is the bisector of ABC. 1. Given.
ABD ≅ DBC 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.
•mABD = mDBC 3. Congruent angles are angles that have the same measure.
Two Column Proof Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements Reasons
Two Column Proof Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements Reasons
1.M is the midpoint of AMB. 1. Given.
Two Column Proof Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements Reasons
1.M is the midpoint of AMB. 1. Given.
2.AM ≅ MB 2. The midpoint of a line segment is the point of that line segment that
divides the segment into congruent segments.
Two Column Proof Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: M is the midpoint of AMB.
Prove: AM = MB
Proof:
Statements Reasons
1.M is the midpoint of AMB. 1. Given.
2.AM ≅ MB 2. The midpoint of a line segment is the point of that line segment that
divides the segment into congruent segments.
3.AM = MB 3. Congruent segments are segments that have the same measure.
Direct and Indirect Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Direct Proof
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Indirect Proof
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Indirect Proof
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be the given and q be the conclusion
• Assume that the negation of the conclusion (~q) is true
• 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
• Since the assumption leads to a condiction, it (~q)must be false. The negation of the assumption (q), the desired conclusion, must be true
Direct Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
Direct Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
1.mCDE ≠ 90 1. Given.
Direct Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
1.mCDE ≠ 90 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.
Direct Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
1.mCDE ≠ 90 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 3. If two intersecting lines do not form DE right angles, then they are not
perpendicular.
Indirect Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
Indirect Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
1.CD is perpendicular to DE 1. Assumption.
Indirect Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
1.CD is perpendicular to DE 1. Assumption.
2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles.
Indirect Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
1.CD is perpendicular to DE 1. Assumption.
2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles.
3.mCDE = 90. 3. If an angle is a right angle, then its degree measure is 90
Indirect Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
•CD is perpendicular to DE 1. Assumption.
CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles.
•mCDE = 90. 3. If an angle is a right angle, then its degree measure is 90
•mCDE ≠ 90 4. Given
Indirect Proof Example
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: mCDE ≠ 90
Prove: CD is not perpendicular to DE
Proof:
Statements Reasons
•CD is perpendicular to DE 1. Assumption.
CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles.
•mCDE = 90. 3. If an angle is a right angle, then its degree measure is 90
•mCDE ≠ 90 4. Given
•CD is not perpendicular to 5. Contradiction in 3 and 4. Therefore, DE the assumption is false and its
negation is true.
Postulates, Theorems, and Proof
ERHS Math Geometry
Mr. Chin-Sung Lin
Postulate (or Axiom)
ERHS Math Geometry
Mr. Chin-Sung Lin
A postulate (or axiom) is a statement whose truth is accepted without proof
Theorem
ERHS Math Geometry
Mr. Chin-Sung Lin
A theorem is a statement that is proved by deductive reasoning
Theorems and Geometry
ERHS Math Geometry
Mr. Chin-Sung Lin
undefined terms
defined terms
postulates
theorems
applications
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Basic Properties of Equality
• Reflexive Property
• Symmetric Property
• Transitive Property
Substitution Postulate
Partition Postulate
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Addition Postulate
Subtraction Postulate
Multiplication Postulate
Division Postulate
Power Postulate
Roots Postulate
Reflexive Property of Equality
ERHS Math Geometry
Mr. Chin-Sung Lin
A quantity is equal to itself
a = a
Algebraic example:
x = x
Reflexive Property of Equality
ERHS Math Geometry
Mr. Chin-Sung Lin
Geometric example:
The length of a segment is equal to itself
AB = ABA B
Symmetric Property of Equality
ERHS Math Geometry
Mr. Chin-Sung Lin
An equality may be expressed in either order
If a = b, then b = a
Algebraic example:
x = 5
then
5 = x
Symmetric Property of Equality
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Transitive Property of Equality
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Transitive Property of Equality
ERHS Math Geometry
Mr. Chin-Sung Lin
Geometric example:
If the lengths of segments are equal to the length of the same segment, they are equal to each other
AB = EF and EF = CD
then
AB = CD
A B
E F
C D
Substitution Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Substitution Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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 CB
X Y
Partition Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Partition Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
Geometric example:
The sum of all the parts of a segment is congruent to the whole segment
AB + BC = AC
AB + BC = AC
A CB
Addition Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Addition Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
AB + BC ≅ CD + BC
A CB D
Subtraction Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Subtraction Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
Geometric example:
If a segment is subtracted from two congruent segments, the differences are congruent
AC ≅ BD and
BC ≅ BC
then
AC - BC ≅ BD - BC
A CB D
Multiplication Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Multiplication Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Division Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Division Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Powers Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
The squares of equal quantities are equal
If a = b, and a2 = b2
Algebraic example:
x = 10
then
x2 = 100
Powers Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
Geometric example:
If the lengths of two hypotenuses are equal, their powers are equal
AB = XY
then
AB2 = XY2
A
BC
X
YZ
Root Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Root Postulate
ERHS Math Geometry
Mr. Chin-Sung Lin
Geometric example:
If the squares of the lengths of two hypotenuses are equal, their square roots are equal
AB2 = XY2
then
AB = XY
A
BC
X
YZ
Identify Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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?
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Which postulate tells us that the measure of an angle is equal to itself?
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Which postulate tells us that the measure of an angle is equal to itself?
Reflexive Property
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If BD is equal to AE, and AE is equal to EC, how do we know that BD is equal to EC?
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If BD is equal to AE, and AE is equal to EC, how do we know that BD is equal to EC?
Transitive Property
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
How do we know that BAF + FAC is equal to BAC?
B
AC
F
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
How do we know that BAF + FAC is equal to BAC?
B
AC
F
Partition Postulate
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If BA is equal to FA, and EC is equal to AD, how do we know BA / EC = FA / AD?
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If BA is equal to FA, and EC is equal to AD, how do we know BA / EC = FA / AD?
Division Postulate
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If AF is equal to AC, how do we know that AF - BD = AC - BD?
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If AF is equal to AC, how do we know that AF - BD = AC - BD?
Subtraction Postulate
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Addition Postulate
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
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If AB BC and LM MN, prove mABC = mLMN
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If AB BC and LM MN, prove mABC = mLMN
Given: AB BC and LM MN
Prove: mABC = mLMN
A
B C
L
M N
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB BC and LM MN
Prove: mABC = mLMN
Proof:
Statements Reasons
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB BC and LM MN
Prove: mABC = mLMN
Proof:
Statements Reasons
•AB BC and LM MN 1. Given.
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB BC and LM MN
Prove: mABC = mLMN
Proof:
Statements Reasons
•AB BC and LM MN 1. Given.
ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right
angles.
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB BC and LM MN
Prove: mABC = mLMN
Proof:
Statements Reasons
•AB BC and LM MN 1. Given.
ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right
angles.
3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB BC and LM MN
Prove: mABC = mLMN
Proof:
Statements Reasons
•AB BC and LM MN 1. Given.
ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right
angles.
3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90
•90 = mLMN 4. Symmetric property of equality
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB BC and LM MN
Prove: mABC = mLMN
Proof:
Statements Reasons
•AB BC and LM MN 1. Given.
ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right
angles.
3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90
4. 90 = mLMN 4. Symmetric property of equality
5. mABC = mLMN 5. Transitive property of equality
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If AB = 2 CD, and CD = XY, prove AB = 2 XY
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If AB = 2 CD, and CD = XY, prove AB = 2 XY
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
A CB D
X Y
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
Proof:
Statements Reasons
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AB = 2 CD, and CD = XY
Prove: AB = 2 XY
Proof:
Statements Reasons
1.AB = 2 CD, and CD = XY 1. Given.
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If ABCD are collinear, AB = CD, prove AC = BD
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
If ABCD are collinear, AB = CD, prove AC = BD
Given: ABCD and AB = CD
Prove: AC = BD
A CB D
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements Reasons
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements Reasons
1.AB = CD 1. Given.
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: ABCD and AB = CD
Prove: AC = BD
Proof:
Statements Reasons
1.AB = CD 1. Given.
2.BC = BC 2. Reflexive property of equality
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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
4.AB + BC = AC 4. Partition postulate
CD + BC = BD
Apply Postulates for Proofs
ERHS Math Geometry
Mr. Chin-Sung Lin
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
4.AB + BC = AC 4. Partition postulate
CD + BC = BD
5. AC = BD 5. Substitution postulate
Q & A
ERHS Math Geometry
Mr. Chin-Sung Lin
The End
ERHS Math Geometry
Mr. Chin-Sung Lin
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