Question

write a two - column proof. given: m∠gfi = 131°. prove: m∠efi = 36°. statements: 1. m∠gfi = 131°. 2. m∠gfe + m∠efi = m∠gfi. 3. m∠gfe + m∠efi = 131°. 4. m∠gfe=(8x - 1)°, m∠efi = 3x°. 5. 8x - 1+3x = 131. 6. 11x - 1 = 131. 7. 11x = 132. 8. x = 12. 9. m∠efi = °. reasons: 1. given. 2. angle addition postulate. 3. substitution property, (steps 1, 2). 4. given. 5. substitution property. 6. combine like terms. 7. addition property of equality. 8. division property of equality. 9. substitution property, (steps 4, 8).

Explanation:

Step1: Recall angle - addition

By the Angle Addition Postulate, $\angle GFE+\angle EFI=\angle GFI$.

Step2: Substitute known values

Since $\angle GFI = 131^{\circ}$, we have $\angle GFE+\angle EFI=131^{\circ}$.

Step3: Substitute angle expressions

Given $\angle GFE=(8x - 1)^{\circ}$ and $\angle EFI = 3x^{\circ}$, then $8x-1 + 3x=131$.

Step4: Combine like - terms

Combining $8x$ and $3x$ gives $11x-1 = 131$.

Step5: Add 1 to both sides

Using the Addition Property of Equality, $11x=131 + 1=132$.

Step6: Solve for x

Dividing both sides by 11 (Division Property of Equality), $x = 12$.

Step7: Find $\angle EFI$

Substitute $x = 12$ into the expression for $\angle EFI$. Since $\angle EFI=3x^{\circ}$, then $\angle EFI=3\times12^{\circ}=36^{\circ}$.

Answer:

$36$