proving statements in geometry

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


Fig. 1 Fig. 2 Fig. 3 Fig. 4

Visual Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin


Fig. 1


Visual Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin



Visual Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin


Fig. 1


Visual Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin



Visual Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin


Fig. 1


Visual Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin



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


1 4 7 10 13 16 19

3 3 3 3 3 3

Number Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin


1 ?4 9 16 ? ?

Number Pattern

ERHS Math Geometry

Mr. Chin-Sung Lin


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


Picture

No of Connections 0 1 3 6 ?

1 2 3 ?

Make a Conjecture

ERHS Math Geometry

Mr. Chin-Sung Lin


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


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Conjecture: the value of x2 always greater than the value of x


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


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


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


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:


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



ERHS Math Geometry

Mr. Chin-Sung Lin


• Angles with measure of 90 are right angles If measure of angles is 90, then their are right anglesq p (T)



ERHS Math Geometry

Mr. Chin-Sung Lin


• 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.


ERHS Math Geometry

Mr. Chin-Sung Lin



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.


ERHS Math Geometry

Mr. Chin-Sung Lin



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



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.


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: BD is the bisector of ABC.

Prove: mABD = mDBC

Proof:

Statements Reasons


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABD = mDBC

Proof:

Statements Reasons

1.BD is the bisector of ABC. 1. Given.


ERHS Math Geometry

Mr. Chin-Sung Lin


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.


ERHS Math Geometry

Mr. Chin-Sung Lin


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


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AM = MB

Proof:

Statements Reasons

1.M is the midpoint of AMB. 1. Given.


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AM = MB

Proof:

Statements Reasons


2.AM ≅ MB 2. The midpoint of a line segment is the point of that line segment that

divides the segment into congruent segments.


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AM = MB

Proof:

Statements Reasons


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


Proof:

Statements Reasons

1.mCDE ≠ 90 1. Given.


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


Proof:

Statements Reasons


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.


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


Proof:

Statements Reasons


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


Proof:

Statements Reasons


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


Proof:

Statements Reasons

1.CD is perpendicular to DE 1. Assumption.


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


Proof:

Statements Reasons


2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles.


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


Proof:

Statements Reasons


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: mCDE ≠ 90


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


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


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


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


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


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


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


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


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?


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Which postulate tells us that the measure of an angle is equal to itself?


ERHS Math Geometry

Mr. Chin-Sung Lin

Which postulate tells us that the measure of an angle is equal to itself?

Reflexive Property


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?


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


ERHS Math Geometry

Mr. Chin-Sung Lin

How do we know that BAF + FAC is equal to BAC?

B

AC

F


ERHS Math Geometry

Mr. Chin-Sung Lin

How do we know that BAF + FAC is equal to BAC?

B

AC

F

Partition Postulate


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?


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


ERHS Math Geometry

Mr. Chin-Sung Lin

If AF is equal to AC, how do we know that AF - BD = AC - BD?


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


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


ERHS Math Geometry

Mr. Chin-Sung Lin

If AB BC and LM MN, prove mABC = mLMN


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


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABC = mLMN

Proof:

Statements Reasons


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABC = mLMN

Proof:

Statements Reasons

•AB BC and LM MN 1. Given.


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABC = mLMN

Proof:

Statements Reasons


ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right

angles.


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABC = mLMN

Proof:

Statements Reasons



angles.

3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABC = mLMN

Proof:

Statements Reasons



angles.


•90 = mLMN 4. Symmetric property of equality


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: mABC = mLMN

Proof:

Statements Reasons



angles.


4. 90 = mLMN 4. Symmetric property of equality

5. mABC = mLMN 5. Transitive property of equality


ERHS Math Geometry

Mr. Chin-Sung Lin

If AB = 2 CD, and CD = XY, prove AB = 2 XY


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


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AB = 2 XY

Proof:

Statements Reasons


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AB = 2 XY

Proof:

Statements Reasons

1.AB = 2 CD, and CD = XY 1. Given.


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AB = 2 XY

Proof:

Statements Reasons

1.AB = 2 CD, and CD = XY 1. Given.

2.AB = 2 XY 2. Substitution postulate


ERHS Math Geometry

Mr. Chin-Sung Lin

If ABCD are collinear, AB = CD, prove AC = BD


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


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AC = BD

Proof:

Statements Reasons


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AC = BD

Proof:

Statements Reasons

1.AB = CD 1. Given.


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AC = BD

Proof:

Statements Reasons

1.AB = CD 1. Given.

2.BC = BC 2. Reflexive property of equality


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AC = BD

Proof:

Statements Reasons

1.AB = CD 1. Given.


3.AB + BC = CD + BC 3. Addition postulate


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AC = BD

Proof:

Statements Reasons

1.AB = CD 1. Given.



4.AB + BC = AC 4. Partition postulate

CD + BC = BD


ERHS Math Geometry

Mr. Chin-Sung Lin


Prove: AC = BD

Proof:

Statements Reasons

1.AB = CD 1. Given.



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

proving statements in geometry

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