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. reasons: 1. given. 2. angle addition postulate. 3. substitution property, (steps 1, 2). 4. given. 5.

Explanation:

Step1: Identify angle - sum relationship

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

Step2: Substitute given angle measures

We know $\angle GFI = 131^{\circ}$, $\angle GFE=(8x - 1)^{\circ}$ and $\angle EFI = 3x^{\circ}$. So, $(8x-1)+3x=131$.

Step3: Combine like - terms

$8x+3x-1 = 131$, which simplifies to $11x-1=131$.

Step4: Add 1 to both sides

$11x-1 + 1=131 + 1$, so $11x=132$.

Step5: Solve for x

Divide both sides by 11: $\frac{11x}{11}=\frac{132}{11}$, then $x = 12$.

Step6: Find the measure of $\angle EFI$

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

Answer:

The reason for step 5 is "Substitution Property (using step 4 for the angle - measure expressions in step 3)" and the proof is complete as we have shown that $m\angle EFI = 36^{\circ}$.