Geometry Section 2-6
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Transcript of Geometry Section 2-6
Section 2-6Algebraic Proof
Wednesday, November 6, 13
Essential Questions
How do you use Algebra to write two-column proofs?
How do you use properties of equality to write geometric proofs?
Wednesday, November 6, 13
Vocabulary
1. Algebraic Proof:
2. Two-column Proof:
3. Formal Proof:
Wednesday, November 6, 13
Vocabulary
1. Algebraic Proof: When a series of algebraic steps are used to solve problems and justify steps.
2. Two-column Proof:
3. Formal Proof:
Wednesday, November 6, 13
Vocabulary
1. Algebraic Proof: When a series of algebraic steps are used to solve problems and justify steps.
2. Two-column Proof: A format for a proof where one column provides statements of what is being done, and the second is to justify each statement
3. Formal Proof:
Wednesday, November 6, 13
Vocabulary
1. Algebraic Proof: When a series of algebraic steps are used to solve problems and justify steps.
2. Two-column Proof: A format for a proof where one column provides statements of what is being done, and the second is to justify each statement
3. Formal Proof: Another name for a two-column proof
Wednesday, November 6, 13
Properties of Real Numbers
Wednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Properties of Real NumbersAddition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality: If a = b, then a − c = b − cMultiplication Property of Equality: If a = b, then a × c = b × c
Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ cReflexive Property of Equality: a = a
Symmetric Property of Equality: If a = b, then b = aTransitive Property of Equality: If a = b and b = c, then a = c
Substitution Property of Equality: If a = b, then a may be replaced by b in any equation/expression
Distributive Property: a(b + c) = ab + acWednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18 Addition Property
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
−10a =110
Addition Property
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
−10a =110
Addition Property
Substitution Property
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
−10a =110
Addition Property
Substitution Property −10 −10
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
−10a =110
Addition Property
Substitution Property −10 −10 Division Property
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
−10a =110
Addition Property
Substitution Property −10 −10 Division Property
a = −11
Wednesday, November 6, 13
Example 1Solve the equation. Write a justification for each step.
2(5−3a)−4(a+7)=92 Given
10−6a−4a−28=92 Distributive Property
−10a−18=92 Substitution Property +18 +18
−10a =110
Addition Property
Substitution Property −10 −10 Division Property
a = −11 Substitution Property
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
Statements Reasons
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
Statements Reasons
1.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
1.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
3.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
d − 520
= t3.
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
d − 520
= t3. Division Property
Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
d − 520
= t3. Division Property
4.Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
d − 520
= t3. Division Property
t = d −5
204.Wednesday, November 6, 13
Example 2If the distance d an object travels is given by , d =20t +5
the time t that the object travels is given by t = d −5
20.
Write a two-column proof to verify this conjecture.
d =20t +5
Statements Reasons
Given1.2. d −5=20t Subtraction Property
d − 520
= t3. Division Property
t = d −5
204. Symmetric PropertyWednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
1.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
4.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
4. m∠A = 2(45°)
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
4. m∠A = 2(45°) Substitution Property
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
4. m∠A = 2(45°) Substitution Property
5.
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
4. m∠A = 2(45°) Substitution Property
5. m∠A = 90°
Wednesday, November 6, 13
Example 3If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.Statements Reasons
∠A ≅ ∠B, m∠B = 2m∠C,and m∠C = 45°
1. Given
2. m∠A = m∠B Def. of ≅ angles
3. m∠A = 2m∠C Transitive Property
4. m∠A = 2(45°) Substitution Property
5. m∠A = 90° Substitution Property
Wednesday, November 6, 13
Problem Set
Wednesday, November 6, 13
Problem Set
p. 137 #1-27 odd, 36
“The greatest challenge to any thinker is stating the problem in a way that will allow a solution.” - Bertrand Russell
Wednesday, November 6, 13