1. CONDITIONALS: IF…, THEN….
STANDARDS 1,2,3
What is a CONJECTURE?
CONDITIONALS: IF…, THEN….
CONVERSE OF A CONDITIONAL
NEGATION OF A CONDITIONAL
INVERSE OF A CONDITIONAL
CONTRAPOSITIVE OF A CONDITIONAL
LAW OF DETACHMENT
LAW OF SYLLOGISM
DEDUCTIVE VS INDUCTIVE?
ELEMENTS TO CONSTRUCT PROOFS
GEOMETRIC PROOF 1
GEOMETRIC PROOF 2
GEOMETRIC PROOF 3
GEOMETRIC PROOF 4
GEOMETRIC PROOF 5
END SHOW
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2. Students write geometric proofs, including proofs by contradiction.
Standard 1:
Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
Standard 2:
Students write geometric proofs, including proofs by contradiction.
Standard 3:
Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.
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3. Estándar 1:
Los estudiantes demuestran entendimiento en identificar ejemplos de términos indefinidos, axiomas, teoremas, y razonamientos inductivos y deductivos.
Standard 2:
Los estudiantes escriben pruebas geométricas, incluyendo pruebas por contradicción.
Standard 3:
Los estudiantes construyen y juzgan la validéz de argumentos lógicos y dan contra ejemplos para desaprobar un estatuto.
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4. I will hit the target with this angle and pulling this way…yes!
STANDARDS 1,2,3
I will hit the target with this angle and pulling this way…yes!
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5. STANDARDS 1,2,3 There it goes…!
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6. STANDARDS 1,2,3 Go ahead arrow…!
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7. I didn’t think about the wind!
STANDARDS 1,2,3
WIND
I didn’t think about the wind!
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8. I didn’t think about the wind!
STANDARDS 1,2,3
WIND
I didn’t think about the wind!
A CONJECTURE is an educated guess, and sometimes may be wrong.
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9. They form a square! ? STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: K(1,1), L(1,3), M(3,3), N(3,1) x y 1
Conjecture: ?
They form a square! L M K N
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10. What conjecture may be made from the given information?
STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A B D E
Conjecture:
Point E noncollinear with points A, B, and D.
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11. STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) ?! x y 1
Conjecture: B C
Or… E A D
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12. STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) x y 1
Conjecture: B C
mhh!..or E A D
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13. What conjecture may be made from the given information?
STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) x y 1
Conjecture:
Guah! this also works…? B C E A D
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14. STANDARDS 1,2,3 ?! x y A B D C E
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) ?! x y A B D C E
Conjecture: x y A B D C E ?!
Sometimes we may reach to more than one conjecture!
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15. STANDARDS 1,2,3
Determine the validity of the conjecture and give a counterexample should the conjecture be false.
Given:
Points A, B, C, D B C
Conjecture:
They only form a square. A D
False!
Counterexample:
They could form an isosceles trapezoid as well! B C A D
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16. p q IF…, THEN … p q If p, then q If p, then q STANDARDS 1,2,3
CONDITIONAL STATEMENTS OR CONDITIONALS:
IF…, THEN … p q
Where:
If p, then q
p = hypothesis
q = conclusion
Convert to conditional statements:
HYPOTHESIS
CONCLUSION
Students study to get good grades
If p, then q
If students study, then they get good grades.
HYPOTHESIS
CONCLUSION
Athletes train hard to win competitions. q p
If athletes train hard, then they win competitions.
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17. p q q p IF…, THEN … IF…, THEN … If p, then q If q, then p CONVERSE:
STANDARDS 1,2,3
CONDITIONAL STATEMENTS OR CONDITIONALS:
IF…, THEN … p q
Where:
If p, then q
p = hypothesis
q = conclusion
CONVERSE:
IF…, THEN … q p
Where:
If q, then p
p = conclusion
q = hypothesis
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18. q p p q IF…, THEN … CONVERSE: STANDARDS 1,2,3
Write the CONVERSE of the following conditional:
HYPOTHESIS
CONCLUSION
Athletes train hard to win competitions.
First convert to If…, then… statement q p
If athletes train hard, then they win competitions.
Now get the converse:
IF…, THEN … p q
CONVERSE:
If they win competitions, then athletes train hard.
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19. q p IF…, THEN … If p, then q CONVERSE: STANDARDS 1,2,3
Write the CONVERSE of the following conditional:
HYPOTHESIS
CONCLUSION
Students study to get good grades
First convert to If…, then… statement
If p, then q
If students study, then they get good grades.
Now get the converse:
IF…, THEN … p q
CONVERSE:
If they get good grades, then students study.
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20. A linear pair has adjacent angles.
STANDARDS 1,2,3
Write the converse of the following true statement, and determine if true or false. If it is false, give a counterexample:
A linear pair has adjacent angles.
Explore:
a) Obtain converse
b) Is it true or false?
c) If false find a counterexample
Plan:
Write the given statement as a conditional:
If a linear pair, then angles are adjacent.
Solve:
a) Converse: If angles are adjacent, then they are a linear pair.
b) It is false
35°
55°
c) Counterexample:
Both angles in the figure at the right are adjacent but not a linear pair.
The converse of a true conditional, not necessarily is true.
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21. ~p is “not p” or the negation of p.
STANDARDS 1,2,3
NEGATION:
The negation of a statement is its denial.
~p is “not p” or the negation of p.
An angle is right
An angle is not right p ~p
An angle is not right
An angle is right p ~p
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22. ~p ~q INVERSE: STANDARDS 1,2,3
The inverse of a conditional statement is when both the hypothesis and the conclusion are denied. ~p ~q
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23. a)Writing the conditional in If-Then form:
STANDARDS 1,2,3
For the true conditional: a linear pair has supplementary angles; write the inverse and determine if true or false. If false give a counterexample:
a)Writing the conditional in If-Then form:
If a linear pair, then it has supplementary angles.
HYPOTHESIS p
CONCLUSION q
b) Negating both the hypothesis and the conclusion:
If not a linear pair then it doesn’t have supplementary angles.
INVERSE
Negated HYPOTHESIS ~p
Negated CONCLUSION ~q
c) Is it true?
The inverse of this conditional is false, as shown in the following counterexample: A C B D E
40°
140°
In the figure at the left both angles ABC and EBD aren’t a linear pair but they are supplementary.
140° + 40° = 180°
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24. IF…, THEN … p q If p, then q IF…, THEN … IF…, THEN … ~q ~p q p
STANDARDS 1,2,3
CONTRAPOSITIVE of a conditional statement:
The contrapositive of a conditional statement is the negation of the hypothesis and conclusion of its converse.
IF…, THEN … q p
If p, then q
IF…, THEN … p q
If q, then p
CONVERSE:
IF…, THEN … ~p ~q
If not q, then not p
CONTRAPOSITIVE:
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25. STANDARDS 1,2,3
Find the contrapositive of the true conditional if two points lie in a plane, then the entire line containing those points lies in that plane. Is the contrapositive true or false?
a) converse:
If the entire line containing those points lies in that plane, then the two points lie in a plane.
b) contrapositive:
If the entire line containing those points does not lie in that plane, then the two points do not lie in a plane.
Counterexample:
FALSE.
Line AB containing points A and B doesn’t lie in plane Q, but A and B do lie in plane R. A B R Q
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26. If p q is a true statement and p is true, then q is true.
STANDARDS 1,2,3
LAW OF DETACHMENT
If p q is a true statement and p is true, then q is true.
If two numbers are even, then their sum is a real number is a true conditional, and 4 and 6 are even numbers. Try to reach a logical conclusion using the Law of Detachment.
If two numbers are even, then their sum is a real number p q
p q
is true
4 and 6 are even
is true p
Conclusion?
4 + 6 = 10, 10 is a real number.
is true
By Law of Detachment q
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27. (a) If you read novels, then you like mystery books. p q is true p q
STANDARDS 1,2,3
Determine if statement (c) goes after statements (a) and (b) by the Law of Detachment. If it does not follow, then write invalid [suppose (a) and (b) true]:
(a) If you read novels, then you like mystery books.
p q
is true p q
(b) Juan read a novel.
is true p
(c) He likes mystery books.
Yes, it follows by Law of Detachment. q
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28. (a) If two angles add up to 90° then they are complementary p q
STANDARDS 1,2,3
Determine if statement (c) goes after statements (a) and (b) by the Law of Detachment. If it does not follow, then write invalid [suppose (a) and (b) true]:
(a) If two angles add up to 90° then they are complementary
p q
is true p q
(b) m A + m B = 90°
is true p
(c) A and B are complementary
Yes, it follows by Law of Detachment. q
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29. (a) If two angles are vertical, then they are congruent p q is true p
STANDARDS 1,2,3
Determine if statement (c) goes after statements (a) and (b) by the Law of Detachment. If it does not follow, then write invalid [suppose (a) and (b) true]:
(a) If two angles are vertical, then they are congruent
p q
is true p q
(b) and are vertical.
is true p
(c) and oppose by the vertex.
Invalid q
What should follow to be true?
(c) and are congruent.
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30. p q p r q r p q p r q r STANDARDS 1,2,3 LAW OF SYLLOGISM:
If p q and q r are true conditionals, then p r is true as well.
p q
q r
p r If
is true,
then
is true.
If a vehicle has four wheels, then it is a car
If it is a car, then you can drive it.
Using the Law of Syllogism, what conclusion may be reached?
If the vehicle has four wheels,
then you can drive it. p q p r q r
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31. STANDARDS 1,2,3
Determine if statement (c) follows from statements (a) and (b) by the Law of Syllogism. In case this is not true, write INVALID.
(a) If a mammal, then it has warm blood. p q
(b) If it has warm blood then it drinks milk. q r
(c) If a mammal, then it drinks milk. p r
Yes, by the Law of Syllogism.
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32. p q p q q r q r p r p r STANDARDS 1,2,3
Determine if statement (c) follows from statements (a) and (b) by the Law of Syllogism. In case this is not true, write INVALID.
Each statement could be read as:
(a) A ABC p q
If A, then congruent to ABC p q
(b) ABC is a right angle.
If ABC, then it is a right angle. q r q r
(c) A is a right angle. p r
If A, then it is a right angle. p r
Yes, by the Law of Syllogism.
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33. STANDARDS 1,2,3
Can a conclusion be reached using the Law of Detachment or the Law of Syllogism from (a) and (b)
(a) ABC is an obtuse angle. p q
(b) An obtuse angle is greater than an acute angle. q r
CONCLUSION:
ABC is greater than an acute angle
by Law of Syllogism p r
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34. Logical Reasoning Deductive Reasoning Inductive Reasoning ?
STANDARDS 1,2,3
Deductive Reasoning
Inductive Reasoning
- Uses a set of rules to prove a statement.
- Finding a general rule based on observation of data, patterns,
and past performance.
4x + 2 = 22
Given:
Prove:
x = 5 4 5 3 2 1
Squares
Step
Proof:
4x + 2 = 22
Subtraction Property of Equality
4x = 20
Division Property of Equality ? 7
Substitution Property of Equality
Rule: We add 2 squares per step.
x = 5
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35. PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c:
STANDARDS 1,2,3
ALGEBRAIC REVIEW
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
COMMUTATIVE PROPERTY:
Addition:
a + b = b + a
5 + 7
= 7 + 5
Multiplication:
a b = b a
9 6
= 6 9
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36. PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c:
STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
ASSOCIATIVE PROPERTY:
Addition:
(a + b) + c = a + (b + c)
(3 + 4) +1
= 3 + (4 + 1)
Multiplication:
a b c= a b c =
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37. PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c:
STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
IDENTITY PROPERTY:
Addition:
a + 0 = 0 + a=a
5 + 0
= 0 + 5
= 5
Multiplication:
a 1 = 1 a = a
9 1
= 1 9
= 9
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38. PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c:
STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
INVERSE PROPERTY:
Addition:
a + (-a) = (-a) + a=0
5 + (-5)
= (-5) + 5
= 0 1 5 1 5 =
= 1
If a=0 then
Multiplication:
a = a = 1 1 a 3 5 = 5 3
= 1
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39. PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c:
STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
DISTRIBUTIVE PROPERTY:
Distributive:
a(b+c) = ab + ac
and
(b+c)a = ba + ca
3(5+1) = 3(5) + 3(1)
and
(5+1)3 = 5(3) + 1(3)
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40. Name the property shown at each equation:
STANDARDS 1,2,3
Name the property shown at each equation:
1 45 = 45 a)
Identity property (X) = b)
Commutative property (+)
(-3) + 3 = 0 c)
Inverse property (+)
5(9 +2) = d)
Distributive property
(2 + 1) +b= 2 + (1 + b) e)
Associative property (+)
-34(23) = 23(-34) f)
Commutative property (X)
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41. PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
STANDARDS 1,2,3
PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
SUBSTITUTION PROPERTY OF EQUALITY: If
b=2
and
3b +1=7
If a=b, then a may be replaced by b.
then
3( )+1=7 2
ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY:
For any numbers a, b, and c, if a=b then a+c=b+c and a-c=b-c
10 = 10
22 = 22
16 = 16
17 = 17
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42. PROPERTIES OF EQUALITY
STANDARDS 1,2,3
PROPERTIES OF EQUALITY
MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY:
For any real numbers a, b, and c, if a=b, then a c=b c and if c=0, = a c b
28 = 28 2
15 = 15
30 = 30
4 = 4
36 = 36 3
24 = 24
72 = 72
3 = 3
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43. Deductive Reasoning: Algebra
STANDARDS 1,2,3
FORMAL
INFORMAL
Two column proofs:
Proof:
Statements
Reasons
Given: 4(x + 2) = 2x + 18
Prove: x = 5
4(x + 2) = 2x + 18
4x + 8= 2x + 18
4x = 2x + 10
-2x -2x
2x = 10
(1) given
(1) 4(x + 2) = 2x + 18
x = 5
(2) 4x + 8= 2x + 18
(2) Distributive prop.
(3) 4x = 2x + 10
(3) Subtraction prop. of equality
(4) 2x = 10
(4) Subtraction prop. of equality
(5) x = 5
(5) Division Prop. of equality.
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44. Deductive Reasoning (GEOMETRY)
STANDARDS 1,2,3
Deductive Reasoning (GEOMETRY)
Conjecture - a statement or conditional trying to prove.
Elements to construct proofs:
a) Undefined terms
- Terms that are so obvious that don’t require to be proven.
point, line, etc.
b) Definitions
- Statements defined using other terms.
Triangle is a 3 sided polygon.
c) Axioms (Postulates)
- Statements or properties that don’t need to be proven to be used in proofs.
If two planes intersect their intersection is a line.
d) Theorems
- Statements or properties that require to be proven to be used in proofs.
If two angles form a linear pair, then they are supplementary angles.
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45. PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
STANDARDS 1,2,3
PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
REFLEXIVE PROPERTY OF EQUALITY:
For any real number a, a=a
5=5
-10=-10
SYMMETRIC PROPERTY OF EQUALITY:
For all real numbers a and b, if a=b, then b=a
X=5
5=X
6X-12=8
8=6X-12
9Y -2Y +1= 3X 2
3X= 9Y -2Y+1 2
TRANSITIVE PROPERTY OF EQUALITY:
For all real numbers a, b, and c, if a=b, and b=c then a=c
If X=6 and Y= 6 then
X=Y
If Y=2X+2 and Y=6-3X then
2X+2=6-3X
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46. STANDARDS 1,2,3
Congruence in segments and angles is Reflexive, Symmetric and Transitive:
For all segments and angles, their measures comply with these same properties.
of segments is reflexive.
of s is reflexive LM
ECA
of segments is symmetric.
of s is symmetric KL LM LM KL
BCE
FGH
BCE
FGH
of segments is transitive.
of s is transitive KL LM
BCE
FGH LM AB
FGH
ECA KL AB
BCE
ECA
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47. DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3
Given:
Prove: M L K B A
L is midpoint of KM LM AB KL
Two Column Proof:
Statements
Reasons
(1)
L is midpoint of KM
Given
(2) KL LM
Definition of Midpoint
(3) LM AB
Given
(4) KL AB
of segments is transitive.
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48. DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3B C F D A E B C F D A E
Given:
EFD
is right
Prove:
AFB
CFB
and are complementary.
Two Column Proof:
Statements
Reasons
(1)
EFD
is right
Given
(2) EC AD
Definition of lines
(3)
AFC
is right
lines form 4 right s
(4)
AFC= m
90°
Definition of right s
(5)
AFB + m
AFC
CFB =
addition postulate
(6)
AFB + m
CFB = 90°
Substitution prop. of (=)
(7)
AFB
CFB
and are complementary.
Definition of complementary s
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49. DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3 Given: BH
CE bisects
BCA
FGH
ECA
Prove:
FGH + m
BCD = 180°
2( )
Two Column Proof:
Statements
Reasons
(1)
CE bisects
BCA
Given
(2)
BCE
ECA
Definition of bisector
(3)
BCE= m
ECA
Definition of s
(4)
FGH
ECA
Given
(5)
FGH= m
ECA
Definition of s
(6)
BCE= m
FGH
of s is transitive
(7)
BCE + m
BCD = 180°
ECA +
addition postulate
(8)
FGH + m
BCD = 180°
Substitution prop. of (=)
(9)
FGH + m
BCD = 180°
2( )
Adding like terms
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50. DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3B C A D F E B C A D
Given:
FBD
is right
Prove:
ABF
CBD
and are complementary.
Two Column Proof:
Statements
Reasons
(1)
FBD
is right
Given
(2)
FBD= m
90°
Definition of right s
(3)
FBD + m
CBD = 180°
ABF +
addition postulate
(4)
CBD = 180° m
ABF +
90° +
Substitution prop. of (=)
(5)
ABF + m
CBD = 90°
Subtraction prop. of (=)
(6)
ABF
CBD
and are complementary.
Definition of complementary s
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51. DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3
Given: C A B H G D E F C A B H G D E F
AC and DF are
GE is a transversal
Prove:
GBC
FEH
and are supplementary.
Two Column Proof:
Statements
Reasons
(1)
GE is a transversal
AC and DF are
Given
(2)
GBC
CBE
and are a linear pair
Definition of linear pair
(3)
GBC + m
CBE = 180°
s in a linear pair are supplementary
(4)
CBE
FEH
In lines cut by a transversal CORRESPONDING s are
(5)
CBE= m
FEH
Definition of s
(6)
GBC + m
FEH = 180°
Substitution prop. of (=)
(7)
GBC
FEH
and are supplementary.
Definition of supplementary s
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