Question

write a two - column proof. given: m∠gfi = 141°. prove: m∠efi = 48°. 2. m∠gfe + m∠efi = m∠gfi | 2. angle addition postulate 3. m∠gfe + m∠efi = 141° | 3. substitution property, (steps 1, 2) 4. m∠gfe=(8x - 3)°, m∠efi = 4x° | 4. given 5. 8x - 3+4x = 141 | 5. substitution property 6. 12x - 3 = 141 | 6. combine like terms. 7. 12x = 144 | 7. addition property of equality 8. x = 12 | 8. division property of equality 9. m∠efi = ° | 9. substitution property, (steps 4, 8)

Explanation:

Step1: Recall angle - addition relationship

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

Step2: Substitute given angle measure

Since $\angle GFI = 141^{\circ}$, we have $\angle GFE+\angle EFI=141^{\circ}$.

Step3: Substitute angle expressions

Given $\angle GFE=(8x - 3)^{\circ}$ and $\angle EFI = 4x^{\circ}$, we get the equation $(8x - 3)+4x=141$.

Step4: Combine like - terms

Combining the $x$ terms, $8x+4x=12x$, so the equation becomes $12x-3 = 141$.

Step5: Isolate the variable term

Using the Addition Property of Equality, add 3 to both sides of the equation: $12x-3 + 3=141 + 3$, which simplifies to $12x=144$.

Step6: Solve for $x$

Using the Division Property of Equality, divide both sides of the equation $12x = 144$ by 12: $\frac{12x}{12}=\frac{144}{12}$, so $x = 12$.

Step7: Find the measure of $\angle EFI$

Since $\angle EFI=4x^{\circ}$, substitute $x = 12$ into the expression. Then $\angle EFI=4\times12^{\circ}=48^{\circ}$.

Answer:

$48$