Question

write a two - column proof.
given: m∠gfi = 139°
prove: m∠efi = 36°
statements reasons

1. m∠gfi = 139° 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 postulate

We know that $\angle GFE+\angle EFI=\angle GFI$ by the Angle Addition Postulate. Given $\angle GFI = 139^{\circ}$, so $\angle GFE+\angle EFI=139^{\circ}$. Also, since $\angle GFE=(9x - 5)^{\circ}$ and $\angle EFI = 3x^{\circ}$, we have $(9x-5)+3x=139$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $9x+3x-5 = 139$, which simplifies to $12x-5=139$.

Step3: Add 5 to both sides

$12x-5 + 5=139 + 5$, so $12x=144$.

Step4: Solve for x

Divide both sides by 12: $\frac{12x}{12}=\frac{144}{12}$, so $x = 12$.

Step5: Find the measure of $\angle EFI$

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

Answer:

The proof is completed as above to show that $m\angle EFI = 36^{\circ}$.