Question

write a two - column proof. given: m∠gfi = 127°. prove: m∠efi = 40°. statements: 1. m∠gfi = 127° (1. given); 2. m∠gfe + m∠efi = m∠gfi (2. angle addition postulate); 3. m∠gfe + m∠efi = ° (3. substitution property, (steps 1, 2))

Explanation:

Step1: Apply angle - addition relationship

Since $\angle GFE=(9x - 3)^{\circ}$ and $\angle EFI = 4x^{\circ}$, and by the Angle Addition Postulate $\angle GFE+\angle EFI=\angle GFI$. Also, $\angle GFI = 127^{\circ}$. So, $(9x - 3)+4x=127$.

Step2: Combine like - terms

Combining the $x$ terms on the left - hand side gives $9x+4x-3 = 127$, which simplifies to $13x-3 = 127$.

Step3: Add 3 to both sides

Adding 3 to both sides of the equation $13x-3 = 127$ gives $13x-3 + 3=127 + 3$, so $13x=130$.

Step4: Solve for $x$

Dividing both sides of the equation $13x = 130$ by 13, we get $x=\frac{130}{13}=10$.

Step5: Find $m\angle EFI$

Since $m\angle EFI = 4x^{\circ}$ and $x = 10$, then $m\angle EFI=4\times10^{\circ}=40^{\circ}$.

Answer:

$40^{\circ}$