Download - 2.6 Algebraic Proof
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2.6 Algebraic Proof
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Objectives Use algebra to write two-column
proofs
Use properties of equality in geometry proofs
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ALGEBRAIC PROPERTIES OF EQUALITY
Reflexive Property a = a. Symmetric Property If a = b, then b
= a. 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. Transitive Property of Equality If a = b and b = c,
then a = c. Distributive Property a(b + c) = ab +
ac. Substitution Property of Equality If a = b, then
you may replace b with a in any
expression.
WE USE THE PROPERTIES TO JUSTIFY ALGEBRAIC STEPS AND SOLVE PROBLEMS.
THIS IS DEDUCTIVE REASONING.
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Original equation
Algebraic Steps Properties
Solve
Distributive Property
Substitution Property
Addition Property
Example 1:
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Substitution Property
Division Property
Substitution Property
Answer:
Example 1:
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Original equation
Algebraic Steps Properties
Distributive Property
Substitution Property
Subtraction Property
Solve Your Turn:
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Substitution Property
Division Property
Substitution Property
Answer:
Your Turn:
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Two-Column Proof Two-Column Proof – A proof format
used in geometry in which an argument is presented with two columns, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof.
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Statements ReasonsProof:
Two-Column Proof
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IfWrite a two-column proof. then
Statements ReasonsProof:
1. Given1.
2. 2. Multiplication Property
3. 3. Substitution
4. 4. Subtraction Property
5. 5. Substitution
6. 6. Division Property
7. 7. Substitution
Example 2a:
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Write a two-column proof. If then
1. Given1.
2. Multiplication Property2.
3. Distributive Property3.
4. Subtraction Property4.
5. Substitution5.
6. Addition Property6.
Proof:Statements Reasons
Example 2b:
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Proof:Statements Reasons
8. Division Property8.
9. Substitution9.
7. Substitution7.
Write a two-column proof. If thenExample 2b:
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Write a two-column proof for the following.
a.
Your Turn:
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1. Given
2. Multiplication Property
3. Substitution4. Subtraction Property5. Substitution
6. Division Property
7. Substitution
Proof:Statements Reasons
1.
2.
3. 4. 5.
6.
7.
Your Turn:
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Prove:
b. Given:
Write a two-column proof for the following. Your Turn:
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Proof:Statements Reasons
1. Given
2. Multiplication Property
3. Distributive Property4. Subtraction Property5. Substitution6. Subtraction Property
7. Substitution
1.
2.
3. 4. 5. 6.
7.
Your Turn:
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Geometric Proof Since geometry also uses variables,
numbers, and operations, many of the algebraic properties of equality are true in geometry. For example:Property Segments Angles
Reflexive AB = AB m 1 = m 1
Symmetric If AB = CD, then CD = AB.
If m 1 = m 2, then m 2 = m 1.
Transitive If AB = CD and CD = EF, then
AB = EF.
If m 1 = m 2 and m 2 = m 3, then
m 1 = m 3.
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Read the Test ItemDetermine whether the statements are true based on the given information.
A I only B I and II C I and III D I, II, and III
MULTIPLE- CHOICE TEST ITEM then which of the following is a valid conclusion?IIIIII
If and
Example 3:
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Solve the Test Item
Statement II:Since the order you name the endpoints of a segment is not important, and TS = PR. Thus, Statement II is true.
Statement I:Examine the given information, GH JK ST and . From the definition of congruence of segments, if , then ST RP. You can substitute RP for ST in GH JK ST to get GH JK RP. Thus, Statement I is true.
Example 3:
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Because Statements I and II only are true, choice B is correct.
Answer: B
Statement IIIIf GH JK ST, then . Statement III is not true.
Example 3:
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If and then which of the following is a valid conclusion?I.II.III.
MULTIPLE- CHOICE TEST ITEM
A I only B I and II C I and III D II and III Answer: C
Your Turn:
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SEA LIFE A starfish has five legs. If the length of leg 1 is 22 centimeters, and leg 1 is congruent to leg 2, and leg 2 is congruent to leg 3, prove that leg 3 has length 22 centimeters.
Given:
m leg 1 22 cm
Prove: m leg 3 22 cm
Example 4:
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1. Given1.
2. Transitive Property2.
Proof:Statements Reasons
3. Definition of congruencem leg 1 m leg 33.
4. Givenm leg 1 22 cm4.
5. Transitive Propertym leg 3 22 cm5.
Example 4:
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DRIVING A stop sign as shown below is a regular octagon. If the measure of angle A is 135 and angle A is congruent to angle G, prove that the measure of angle G is 135.
Your Turn:
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Proof:Statements Reasons
1. Given
2. Given
3. Definition of congruent angles
4. Transitive Property
1.
2.
3.
4.
Your Turn:
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Assignment Geometry:
Pg. 97 – 98 #4 – 9, 14 – 25
Pre-AP Geometry:Pg. 97 – 98
#14 – 31