Geometry Definitions, Postulates, and Theorems

Chapter 2: Reasoning and Proof

Section 2.1: Use Inductive Reasoning Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Conjecture - An unproven statement that is based on observations and patterns. Ex. What’s next? x Visual Pattern a) b) c)

x Number Pattern a) 2, 5, 8, 11, ____

b) 5, 7, 11, 17, 25, ____ Inductive Reasoning - Process of looking for patterns and making conjectures. Ex. The product of an odd number and an even number is .

¿¾½

observed? patternthe is What

Ex. The sum of an odd number and an even number is . Counterexample – An example that shows that a conjecture is false. Ex. The difference between two positive numbers is always positive. Find a counterexample. (there are many correct answers)

14

35

add 3 to preceding number

add 2, then 4, then 6, ...

even

3(8) = 24 6(5) = 30

11(24) 264 7(10)= 70

odd

1 + 4 = 5 21 + 10 = 31

8 + 37 = 45 13 + 18 = 31

3 - 9 = -6

Section 2.2: Analyze Conditional Statements Standards: 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Conditional Statement – A logical statement that has two parts, a hypothesis

and a conclusion. A conditional statement is best when written in IF . . . , THEN . . . form.

Hypothesis – The "IF" part of a conditional statement. Conclusion – The "THEN" part of a conditional statement. Ex. Rewrite the conditional statements in if-then form: a) It is time for dinner if it is 6pm. _____________________________________________________ b) All monkeys have tails. ___________________________________________________________ *Conditional statements can be true or false. Decide whether the statement is true or false. If false, provide a counterexample. Ex. If a number is odd, then it is divisible by 3. Converse – A statement formed by switching the hypothesis and conclusion

of a conditional statement. NOTE: A converse is not always a true statement.

Ex. Statement: If you see lightning, then you hear thunder. Converse: _______________________________________________________________________ Negation – Writing the opposite or negative of the statement. Ex. a) $35 �Am , b) D� is obtuse,

Inverse – The opposite of the hypothesis AND the opposite of the conclusion of a conditional statement.

Ex. Statement: If an animal is a dog, then it has four legs. Inverse: ________________________________________________________________________ Contrapositive – The opposite of the hypothesis AND the opposite of the

conclusion of a converse. Ex. Statement: If an animal is a fish, then it can swim. Contrapositive: ___________________________________________________________________ (over)

If it is 6pm, then it is time for dinner.

If an animal is a monkey, then it has a tail.

f you hear thunder, then you see lightening.

m A = 35 D is not obtuse

If an animal is not a dog, then it does not have four legs.

If an animal cannot swim, then it is not a fish.

Equivalent Statements - Statements that are both true or both false. Ex. a) Statement: If o30 �Am , then A� is acute. Contrapositive: ____________________________________________________

b) Inverse: __________________________________________________________

Converse: ________________________________________________________ ***Perpendicular Lines – IF two lines intersect to form a right angle,

THEN the lines are perpendicular.

Biconditional Statement – A statement that contains the phrase "if and only if." This is equivalent to a conditional statement and it's converse.

Biconditional Statement: x + 3 = 5 if and only if x = 2. Conditional: _________________________________________________________________T/F Converse: ___________________________________________________________________T/F The biconditional statement is ___ since the conditional statement is ___ and the converse is ___. Biconditional Statement: x2 = 4x if and only if x = 4. Conditional: _________________________________________________________________T/F Converse: ___________________________________________________________________T/F The biconditional statement is ___ since the conditional statement is ___ and the converse is ___.

If A is not acute, then m A = 30.

If m A = 30, then A is not acute.

If A is acute, then m A = 30both are false

both are true

A

C

B

D

AB CD

notation means perpendicular

If x + 3 = 5, then x = 2.

If x = 2, then x + 3 = 5.

true true true

If x = 4x, then x = 4.

If x = 4, then x = 4x.

false false

2

2

*Can be written as "iff"

Section 2.3: Apply Deductive Reasoning Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Deductive Reasoning – relies on facts, definitions and accepted properties in

logical order to write a logical argument. Inductive Reasoning – examples and patterns are used to form a conjecture. *Two types of Deductive Reasoning:

Law of Detachment – IF p qo is true, AND p is true, THEN q is true.

Ex.

Law of Syllogism – IF p qo is true, AND q ro is true,

THEN p ro is true. Ex. Ex. Decide whether deductive or inductive reasoning is being used in the following argument. Everyday Mrs. Maya reads off the homework answers, then goes over the wrong ones on the board. Today she has read off the homework answers, so you can conclude that she will go over the wrong answers on the board. Why did you decide so? Ex. Law of Syllogism p qo If you are a student, then you have lots of homework.

q ro If you have lots of homework, then you have little social life. p ro .

(over)

If m ABC < 90, then ABC is acute.

Given that m DEF = 42,we can conlude that DEF is acute.

*Like the transitive property (cut out the middle man)

If m ABC = 45, then m ABC < 90.

If m ABC <90, then ABC is acute.

If m ABC = 45, then ABC is acute.

Inductive Based on patterns.

If you are a student, then you have little social life.

Section 2.3 Extension Symbolic Notation: p represents the hypotheses q represents the conclusion conditional statement (if-then): p qo converse q po negation of p ~ p inverse ~ ~p qo contrapositive ~ ~q po biconditional p ql (means p qo and q po ) Ex. Use p and q to write the symbolic statement in words.

p: A number is divisible by 3.

q: A number is divisible by 6. p qo

q po ~ p ~ ~p qo ~ ~q po p ql

If a number is divisible by 3, then it is divisible by 6.

If a number is divisible by 6, then it is divisible by 3.

A number is not divisible by 3.

If a number is not divisible by 3, then it is not divisible by 6.

If a number is not divisible by 6, then it is not divisible by 3.

A number is divisible by 3 if and only if a number is divisible by 6.

Section 2.4: Use Postulates and Diagrams Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Postulate 5 – IF you are given any two points, THEN there exists exactly one line.

Postulate 6 – IF you are given a line, THEN it contains at least two points.

Postulate 7 – IF two lines intersect, THEN their intersection is exactly one point.

Postulate 8 – IF you are given any three noncollinear points, THEN there exists exactly one plane.

Postulate 9 – IF you are given a plane, THEN it contains at least three noncollinear points.

Postulate 10 – IF two points lie in a plane, THEN the line containing them also lies in the plane.

Postulate 11 – IF two planes intersect, THEN the intersection is a line.

Line Perpendicular to a Plane – A line that intersects a plane at a point and makes a right angle with the plane.

Ex. State the postulate that verifies the truth of the statement: V and T lie on line n . Write out the postulate. n

V T

A B

C B U

PR Q

AB C

A B

M

R

A and B are on line

Line contains C & B

Lines and intersect at point P.

P, Q, and R are on plane . (or plane PRQ)

Plane contains points A, B, and c.

A and B are on plane , then AB lies on plane .AB

MR

Planes and intersect at line .M R

S

T

ST plane*a plane has infinitely many lines

Postulate 5: Through any two points, there is exactly one line.

Section 2.5: Reason Using Properties from Algebra Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.

Addition Property of Equality – IF a = b, THEN a + c = b + c.

Subtraction Property of Equality – IF a = b, THEN a – c = b – c.

Multiplication Property of Equality – IF a = b, THEN ac = bc.

Division Property of Equality – IF a = b, THEN a y c = b y c.

***Substitution Property of Equality – IF a = b, THEN a can be substituted for b anytime.

Distributive Property – acabcba � � )( OR )( cbaacab � � acabcba � � )( OR )( cbaacab � �

Properties of Equality for Real Numbers, Segments and for Angles

Real Numbers Segments Angles ***Reflexive IF a is a real # IF Segment AB, IF Angle A, THEN a = a THEN AB = AB. THEN AmAm � � . ***Symmetric IF a = b IF CDAB , IF BmAm � � ,

THEN b = a THEN ABCD . THEN AmBm � � . ***Transitive IF a = b IF CDAB , IF BmAm � � ,

AND b = c, AND E FCD , AND CmBm � � , THEN a = c. THEN EFAB . THEN CmAm � � .

(over)

x - 2 = 10 x = 12

5 + n = 8 n = 3

x/2 = 4 x = 8

4y = 20 y = 5

If m A = 100, and m A = m B, then m B = 100.

add 2 to both sides

subtract 5 from both sides

multiply both sides by 2

Divide both sides by 4

2(x + 4) = 2x + 8 or 5x - 3x = x(5x - 3)

10 = 10 x = x

If 10 = x, then x = 10.

If a = x + 3, and x + 3 = 7, then a = 7.

*cut out the middle man

Ex. Algebraic Proof – Solve the equation 188123 � � xx and state the reason for each step. Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Ex. Geometric Proof – Show the perimeter of ABC' is equal to the perimeter of ADC' . Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Ex. Use the property to complete the statement. Transitive Property: If DBYZ and JK , then

A

B C D

3x + 12 = 8x - 18

12 = 5x - 18

30 = 5x

6 = x

x = 6

Given

Subtraction Prop. of =

Addition Prop. of =

Division Prop. of =

Symmetric Prop.

AD = AB; DC = BC

AC = ACAB + BC + CA = Perimeter of ABC AD + DC + CA = Perimeter of ADC

AD + DC + CA = Perimeter of ABC

Perimetr of ADC = Perimetr of ABC

Given

Reflexive

Definition of Perimeter

Substitution

Substitution

DB YZ = JK

Section 2.6 – Prove Statements about Segments & Angles Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2.0 Students write geometric proofs, including proofs by contradiction.

Proof – A logical argument that shows a statement is true. Two-Column Proof – Numbered statements and corresponding reasons

that show an argument in a logical order.

Statements Reasons 1) 1) 2) 2) etc) etc)

Theorem – A statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs.

***Theorem 2.1 Congruence of Segments is Reflexive GIVEN: Segment AB,

THEN AB AB# .

***Theorem 2.1 Congruence of Segments is Symmetric IF AB CD# , THEN CD AB# .

***Theorem 2.1 Congruence of Segments is Transitive IF AB CD# , AND CD EF# , THEN AB EF# .

Ex. Given: XYPQ # P Q Prove: PQXY X Y 1. 1. Given

2. PQ = XY 2.

3. 3. Symmetric Property

(over)

Statement Reason

PQ = XYDef. of = Segments

XY = PQ

Ex. Given: Q is the midpoint of PR . P Q R

Prove: PQ = PR21

1. Q is the midpoint of PQ 1.

2. 2. Definition of Midpoint

3.PR = PQ +QR 3.

4. 4. Substitution

5. 12

PR = PQ 5.

6. 6.

Ex. Given: E F = GH Prove: EG = F H E F G H 1. 1. Given

2. FG = FG 2.

3. EF+FG=GH+FG 3.

4. 4. Segment Addition Postulate

5. EG = FH 5.

Ex. Given: 12, # BHAHBH Prove: 12 AH 1. 12, # BHAHBH 1. Given

2. 2. Definition of Congruent Segments

3. 3.

4. AH = 12 4.

Ex. In the diagram, if CDBCandBCAB ## , find BC.

B C

A D

H

A D

B C

13 �x 32 �x

GivenPQ = QR

Seg. Add. Post.PR = PQ +PQ, PR = 2PQ

Division Prop. of =PQ = PR Symmetric Prop.

EF = GHReflexiveAdd. Prop. of =

EG = EF + FG,

SubstitutionFH = GH + FG

*Combine like terms

BH = AH12 = AH

12

SubstitutionSymmetric Prop.

3x - 1 = 2x + 3x - 1 = 3

x = 4

BC = 2x + 3BC = 2( ) + 34BC = 11

(over)

***Theorem 2.2 Congruence of Angles is Reflexive GIVEN: Angle A,

THEN AA �#� .

***Theorem 2.2 Congruence of Angles is Symmetric IF BA �#� , THEN AB �#� .

***Theorem 2.2 Congruence of Angles is Transitive

IF BA �#� , AND CB �#� , THEN CA �#� .

Ex. Proof of Transitive Property of Congruence Given: CBBA �#��#� , Prove: CA �#� A B C Statements Reasons 1) CBBA �#��#� , 1) given 2) CmBmBmAm � �� � , 2) 3) CmAm � � 3) 4) CA �#� 4) Ex. Given: 32,43,21 �#��#��#� Prove: 41 �#�

1. 32,43,21 �#��#��#� 1. 2. �1 # �3 2. 3. 3.

Ex. Given: 43,31,631 �#��#�q �m Prove: q � 634m

1. 43,31,631 �#��#�q �m 1. 2. 2. Transitive Property 3. m�1 = m�4 3. 4. 4. Substitution 5. 5.

1 2

3 4

1 2 3 4

Def. of = anglesTransitive PropertyDef. of = angles

Given

1 = 4TransitiveTransitive

Given1 = 4

Def. of = angles63 = m 4m 4 = 63 Symmetric

Section 2.7 – Prove Angle Pair Relationships Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2.0 Students write geometric proofs, including proofs by contradiction. ***Theorem 2.3 Right Angle Congruence Theorem All right angles are congruent. Ex. Given: ABCandDAB �� are right angles, BCDABC �#� Prove: BCDDAB �#�

1. ABCandDAB �� are right angles 1. BCDABC �#�

2. 2. All right triangles are congruent 3. BCDDAB �#� 3.

***Theorem 2.4 Congruent Supplements Theorem IF two angles are supplementary to the same angle (or # angles), THEN they are congruent. ***Theorem 2.5 Congruent Complements Theorem IF two angles are complementary to the same angle (or # angles), THEN they are congruent.

Ex. Given: arycomplementareand

arycomplementareandmm43

,21,243,241��

��q �q �

Prove: 42 �#�

1. q �q � 243,241 mm 1.

arycomplementareandarycomplementareand

43,21

����

2. 2. Transitive Property 3. 31 �#� 3. 4. 42 �#� 4. Substitution

***Linear Pair Postulate – IF two angles form a linear pair, Then they are supplementary o18021 ��� (over)

1 2

A B

C D

1 2 3 4

Given

DAB = ABCTransitive

Given

m 1 = m 3Def. of = anglesCongruent Comp. Theorom.

***Theorem 2.6 Vertical Angles Theorem Vertical angles are congruent. Ex. Given: 21 �� and are a linear pair, 32 �� and are a linear pair Prove: 31 �#�

1. 21 �� and are a linear pair 1. 32 �� and are a linear pair

2. 2. Linear Pair Postulate

3. 31 �#� 3. Ex. See diagram q � � 90CQEmBQDm a) �AQGm b) �CQAm c) If �q � EQFmthenCQDm ,31 d) If �q � CQFmthenBQGm ,125 e) ����� EQGmGQFmAQBm f) If �q � BQCmthenEQFm ,38

1 2 3

A

B

C

D

Q E

G F

Given

1 & 2 are supp. 2 & 3 are supp.

Congruent Supp. Theorem

90 (vertical angles)

90 ( linear pair)

31

125

180

52