Question

write a two - column proof. given: m∠gfh = 127°. prove: m∠efh = 33°. statements: 1. m∠gfh = 127°. 2. m∠gfe + m∠efh = m∠gfh. 3. m∠gfe + m∠efh = 127°. 4. m∠gfe=(9x - 5)°, m∠efh = 3x°. 5. 9x - 5+3x = 127.

Explanation:

Step1: Recall angle - addition postulate

We know that $\angle GFE+\angle EFI=\angle GFI$ (by the angle - addition postulate). Given $\angle GFI = 127^{\circ}$, so $m\angle GFE + m\angle EFI=127^{\circ}$.

Step2: Substitute angle measures

Given $m\angle GFE=(9x - 5)^{\circ}$ and $m\angle EFI = 3x^{\circ}$, we substitute these into the equation $m\angle GFE + m\angle EFI=127^{\circ}$ to get $(9x - 5)+3x=127$.

Step3: Combine like - terms

Combining the $x$ terms on the left - hand side, we have $9x+3x-5 = 127$, which simplifies to $12x-5 = 127$.

Step4: Use the addition property of equality

Add 5 to both sides of the equation $12x-5 = 127$. We get $12x-5 + 5=127 + 5$, which simplifies to $12x=132$.

Step5: Use the division property of equality

Divide both sides of the equation $12x=132$ by 12. So, $\frac{12x}{12}=\frac{132}{12}$, and $x = 11$.

Step6: Find the measure of $\angle EFI$

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

Answer:

$m\angle EFI = 33^{\circ}$