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