Question

write a two - column proof. given: m∠gfi = 131°. prove: m∠efi = 33°. statements: 1. m∠gfi = 131°. reasons: 1. given. 2. m∠gfe + m∠efi = m∠gfi. 2. options: addition property of equality, definition of angle bisector, angle addition postulate

Explanation:

Step1: Identify the angle - addition relationship

By the Angle Addition Postulate, the measure of the whole angle $\angle GFI$ is equal to the sum of the measures of its non - overlapping sub - angles $\angle GFE$ and $\angle EFI$, so $m\angle GFE + m\angle EFI=m\angle GFI$.
We know that $m\angle GFE=(9x - 1)^{\circ}$ and $m\angle EFI = 3x^{\circ}$ and $m\angle GFI=131^{\circ}$. So, $(9x - 1)+3x=131$.

Step2: Combine like terms

Combine the $x$ terms on the left - hand side: $9x+3x-1 = 131$, which simplifies to $12x-1 = 131$.

Step3: Add 1 to both sides

Using the Addition Property of Equality, add 1 to both sides of the equation: $12x-1 + 1=131 + 1$, resulting in $12x=132$.

Step4: Solve for x

Divide both sides of the equation $12x = 132$ by 12 (using the Division Property of Equality): $x=\frac{132}{12}=11$.

Step5: Find the measure of $\angle EFI$

Since $m\angle EFI = 3x^{\circ}$, substitute $x = 11$ into the expression. So, $m\angle EFI=3\times11^{\circ}=33^{\circ}$.

Answer:

The reason for $m\angle GFE + m\angle EFI=m\angle GFI$ is the Angle Addition Postulate. And we have proven that $m\angle EFI = 33^{\circ}$.