5. Reasoning in Algebra and Geometry
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Review the idea of a two-column proof. Remember to list each step and explain your reasoning.
See solution.
Let's begin with reviewing the idea of a two-column proof. It lists each statement on the left and the justification is on the right. Each statement must follow logically from the steps before it. In this case we are given that m ∠ GFI=128, and this how we will begin our proof! Statement 1)& m ∠ GFI=128 Reason 1)& Given From the graph we can notice that the point E is in the interior of ∠ GFI. Thus, from the Angle Addition Postulate we can conclude that m ∠ GFE+ m ∠ EFI = m ∠ GFI. Statement 2)& m ∠ GFE+ m ∠ EFI= m ∠ GFI Reason 2)& Angle Addition Postulate Next, we can tell that the measure m ∠ GFE is 9x-2, and the measure m ∠ EFI is 4x. Also, we are given that m ∠ GFI=128, so let's substitute these expressions in our equation.
| Equation | Substitute values |
|---|---|
| m ∠ GFE+m ∠ EFI =m ∠ GFI | 9x-2+4x=128 |
We could substitute by the means of Substitution Property of Equality. Statement 3)& 9x-2+4x=128 Reason 3)& Substitution Prop. of Equality To isolate x in our equation, we can begin by combining like terms. To do it we can use the Distributive Property and rewrite the equation as shown below. 9 x-2+4 x=128 ⇔ (9+4) x -2=128 After adding the terms in the parentheses, our equation becomes 13x-2=128. Statement 4)& 13x-2=128 Reason 4)& Distributive Property Next, to isolate 13x on one side, we can use the Addition Property of Equality. We add 2 to both sides of the equation, giving us 13x=130. Statement 5)& 13x=130 Reason 5)& Addition Prop. of Equality Now we can use the Division Property of Equality and divide both sides of the equation by 13. This will isolate x. Statement 6)& x=10 Reason 6)& Division Property of Equality We want to calculate m ∠ EFI. Remember that we are given its equivalent expression 4x. Let's list this as the next step in our proof, as we will use this to find m ∠ EFI. Statement 7)& m ∠ EFI = 4x Reason 7)& Given To find this measure, we can substitute x=10 in the expression by the means of the Substitution Property of Equality. Statement 8)& m ∠ EFI = 4(10) Reason 8)& Substitution Prop. of Equality We can solve this expression by multiplying! m ∠ EFI = 4(10) ⇔ m ∠ EFI= 40 We proved that m ∠ EFI=40. Finally, we can complete our two-column table!
Statements
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Reasons
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1. m ∠ GFI=128
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1. Given
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2. m ∠ GFE+ m ∠ EFI = m ∠ GFI
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2. Angle addition Postulate
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3. 9x-2+4x=128
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3. Substitution Property of Equality
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4. 13x-2=128
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4. Distributive Property
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5. 13x=130
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5. Addition Property of Equality
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6. x=10
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6. Division Property of Equality
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7. m ∠ EFI = 4x
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7. Given
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8. m ∠ EFI = 4(10)
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8. Substitution Property of Equality
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9. m ∠ EFI = 40
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9. Multiply
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