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Question
write a two - column proof. given: m∠gfi = 143°. prove: m∠efi = 36°. statements: 1. m∠gfi = 143°. reason: 1. given. 2. m∠gfe + m∠efi = m∠gfi. reason: 2. dropdown with options: addition property of equality, definition of angle bisector, angle addition postulate
Step1: Identify angle - addition relationship
By the Angle Addition Postulate, if we have two adjacent angles $\angle GFE$ and $\angle EFI$ that together form $\angle GFI$, then $m\angle GFE + m\angle EFI=m\angle GFI$.
Step2: Substitute angle - measures
We know that $m\angle GFE=(9x - 1)^{\circ}$, $m\angle EFI = 3x^{\circ}$, and $m\angle GFI = 143^{\circ}$. So, $(9x - 1)+3x=143$.
Step3: Combine like - terms
Combining the $x$ terms on the left - hand side gives $9x+3x-1 = 143$, which simplifies to $12x-1 = 143$.
Step4: Solve for $x$
Add 1 to both sides: $12x-1 + 1=143 + 1$, so $12x=144$. Then divide both sides by 12: $x=\frac{144}{12}=12$.
Step5: Find $m\angle EFI$
Since $m\angle EFI = 3x^{\circ}$, substitute $x = 12$ into the expression. So, $m\angle EFI=3\times12^{\circ}=36^{\circ}$.
For the two - column proof:
| Statements | Reasons |
|---|---|
| 2. $m\angle GFE+m\angle EFI=m\angle GFI$ | 2. Angle Addition Postulate |
| 3. $m\angle GFE=(9x - 1)^{\circ}$, $m\angle EFI = 3x^{\circ}$ | 3. Given (from the angle - labeling in the diagram) |
| 4. $(9x - 1)+3x=143$ | 4. Substitution Property of Equality |
| 5. $12x-1 = 143$ | 5. Combining like terms |
| 6. $12x=144$ | 6. Addition Property of Equality |
| 7. $x = 12$ | 7. Division Property of Equality |
| 8. $m\angle EFI=3x^{\circ}$ | 8. Given |
| 9. $m\angle EFI=3\times12^{\circ}=36^{\circ}$ | 9. Substitution Property of Equality |
The answer for the blank in the two - column proof (reason for statement 2) is: Angle Addition Postulate.
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Step1: Identify angle - addition relationship
By the Angle Addition Postulate, if we have two adjacent angles $\angle GFE$ and $\angle EFI$ that together form $\angle GFI$, then $m\angle GFE + m\angle EFI=m\angle GFI$.
Step2: Substitute angle - measures
We know that $m\angle GFE=(9x - 1)^{\circ}$, $m\angle EFI = 3x^{\circ}$, and $m\angle GFI = 143^{\circ}$. So, $(9x - 1)+3x=143$.
Step3: Combine like - terms
Combining the $x$ terms on the left - hand side gives $9x+3x-1 = 143$, which simplifies to $12x-1 = 143$.
Step4: Solve for $x$
Add 1 to both sides: $12x-1 + 1=143 + 1$, so $12x=144$. Then divide both sides by 12: $x=\frac{144}{12}=12$.
Step5: Find $m\angle EFI$
Since $m\angle EFI = 3x^{\circ}$, substitute $x = 12$ into the expression. So, $m\angle EFI=3\times12^{\circ}=36^{\circ}$.
For the two - column proof:
| Statements | Reasons |
|---|---|
| 2. $m\angle GFE+m\angle EFI=m\angle GFI$ | 2. Angle Addition Postulate |
| 3. $m\angle GFE=(9x - 1)^{\circ}$, $m\angle EFI = 3x^{\circ}$ | 3. Given (from the angle - labeling in the diagram) |
| 4. $(9x - 1)+3x=143$ | 4. Substitution Property of Equality |
| 5. $12x-1 = 143$ | 5. Combining like terms |
| 6. $12x=144$ | 6. Addition Property of Equality |
| 7. $x = 12$ | 7. Division Property of Equality |
| 8. $m\angle EFI=3x^{\circ}$ | 8. Given |
| 9. $m\angle EFI=3\times12^{\circ}=36^{\circ}$ | 9. Substitution Property of Equality |
The answer for the blank in the two - column proof (reason for statement 2) is: Angle Addition Postulate.