Download Geometry Definitions, Postulates, and Theorems

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 Ex. What’s next?
 Visual Pattern
a)
b)
c)
 Number Pattern a)
b)
Inductive Reasoning Ex. The product of an odd number and an even number is
.
What is the


pattern observed? 
Ex. The sum of an odd number and an even number is
.
Counterexample –
Ex. The difference between two positive numbers is always positive. Find a counterexample.
(there are many correct answers)
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 –
Hypothesis –
Conclusion –
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 –
Ex. Statement: If you see lightening, then you hear thunder.
Converse:
Negation –
Ex. a) mA  35 ,
b) D is obtuse,
Inverse –
Ex. Statement: If an animal is a dog, then it has four legs.
Inverse:
Contrapositive –
Ex. Statement: If an animal is a fish, then it can swim.
Contrapositive:
Equivalent Statements Ex. a) Statement: If mA  30 o , then  A is acute.
Contrapositive:
b) Inverse:
Converse:
***Perpendicular Lines –
Biconditional Statement –
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 –
Inductive Reasoning –
*Two types of Deductive Reasoning:
Law of Detachment –
Ex.
Law of Syllogism –
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  q If you are a student, then you have lots of homework.
q  r If you have lots of homework, then you have little social life.
pr
.
Section 2.3 Extension
Symbolic Notation: p represents the hypotheses
q represents the conclusion
Ex.
conditional statement (if-then):
pq
converse
q p
negation of p
~ p
inverse
~ p ~ q
contrapositive
~ q ~ p
biconditional
p  q (means p  q and q  p )
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q
q p
~ p
~ p ~ q
~ q ~ p
pq
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 –
Postulate 6 –
Postulate 7 –
Postulate 8 –
Postulate 9 –
Postulate 10 –
Postulate 11 –
Line Perpendicular to a Plane –
Ex. State the postulate that verifies the truth of the statement: V and T lie on line n .
n
V
T
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  c = b  c.
***Substitution Property of Equality – IF a = b,
THEN a can be substituted for b anytime.
Distributive Property –
a (b  c)  ab  ac OR ab  ac  a (b  c)
a (b  c)  ab  ac OR ab  ac  a (b  c)
Properties of Equality for Real Numbers, Segments and for Angles
Real Numbers
IF a is a real #
THEN a = a
Segments
IF Segment AB,
THEN AB = AB.
Angles
IF Angle A,
THEN mA  mA .
***Symmetric
IF a = b
THEN b = a
IF AB  CD ,
THEN CD  AB .
IF mA  mB ,
THEN mB  mA .
***Transitive
IF a = b
AND b = c,
THEN a = c.
IF AB  CD ,
AND CD  EF ,
THEN AB  EF .
IF mA  mB ,
AND mB  mC ,
THEN mA  mC .
***Reflexive
(over)
Ex. Algebraic Proof – Solve the equation 3x  12  8x 18 and state the reason for each step.
A
B
Ex. Geometric Proof –
C
Given: mAEB  mCED
Prove: mAEC  mBED
E
Ex. Use the property to complete the statement.
Transitive Property: If YZ  DB and
 JK , then
D
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 –
Two-Column Proof –
Statements
Reasons
1)
1)
2)
2)
etc)
etc)
Theorem –
***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.
1.
2.
3.
Given: PQ  XY
Prove: XY  PQ
P
X
Q
Y
(over)
Ex.
Given: Q is the midpoint of PR .
1
Prove: PQ = PR
2
P
Q
R
1.
2.
3.
4.
5.
6.
Ex.
Given: EF = GH
Prove: EG = FH
E
F
G
H
1.
2.
3.
4.
5.
Ex.
B
Given: BH  AH , BH  12
Prove: AH  12
C
H
D
A
1.
2.
3.
4.
Ex. In the diagram, if AB  BC and BC  CD , find BC.
A
D
3x 1
2x  3
B
C
***Theorem 2.2
Congruence of Angles is Reflexive
GIVEN: Angle A,
THEN A  A .
***Theorem 2.2
Congruence of Angles is Symmetric
IF A  B ,
THEN B  A .
***Theorem 2.2
Congruence of Angles is Transitive
IF A  B ,
AND B  C ,
THEN A  C .
Ex.
Proof of Transitive Property of Congruence
Given: A  B, B  C
Prove: A  C
Statements
1) A  B, B  C
2) mA  mB, mB  mC
3) mA  mC
4) A  C
Ex.
Ex.
A
B
C
Reasons
1) given
2)
3)
4)
Given: 1  2, 3  4, 2  3
Prove: 1  4
Given: m1  63, 1  3, 3  4
Prove: m4  63
1
3
4
2
1
3
2
4
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
Ex.
Ex.
Right Angle Congruence Theorem
Given: DAB and ABC are right angles, ABC  BCD
Prove: DAB  BCD
***Theorem 2.4
Congruent Supplements Theorem
***Theorem 2.5
Congruent Complements Theorem
m1  24, m3  24, 1 and 2 are complementary ,
3 and 4 are complementary
Prove: 2  4
D
C
A
B
Given:
***Linear Pair Postulate –
1  2  180 o
1
2
1 2
3 4
***Theorem 2.6
Vertical Angles Theorem
Vertical angles are congruent.
Ex.
Given: 1 and 2 are a linear pair, 2 and 3 are a linear pair
Prove: 1  3
Ex. See diagram
1
3 2
mBQD  mCQE  90
a) mAQG 
C
B
b) mCQA 
D
c) If mCQD  31, then mEQF 
d) If mBQG  125, then mCQF 
A
Q
E
e) mAQB  mGQF  mEQG 
f) If mEQF  38, then mBQC 
F
G