Question

write a two - column proof. given: m∠gfi = 127°. prove: m∠efi = 44°. 1. m∠gfi = 127° 1. 2. m∠gfe + m∠efi = m∠gfi 2. 3. m∠gfe + m∠efi = 127° 3. 4. m∠gfe=(8x - 5)°, m∠efi = 4x° 4. 5. 8x - 5+4x = 127 5.

Explanation:

Step1: Recall angle - addition postulate

By the angle - addition postulate, the measure of the whole angle is the sum of the measures of its non - overlapping parts. So, $\angle GFE+\angle EFI=\angle GFI$.

Step2: Substitute given values

Since $\angle GFI = 127^{\circ}$, we substitute this value into the equation from step 1, getting $\angle GFE+\angle EFI=127^{\circ}$.

Step3: Substitute expressions for angles

We are given that $\angle GFE=(8x - 5)^{\circ}$ and $\angle EFI = 4x^{\circ}$. Substituting these into the equation $\angle GFE+\angle EFI=127^{\circ}$, we get $(8x - 5)+4x=127$.

Step4: Combine like terms

Combining the $x$ terms on the left - hand side, we have $(8x+4x)-5 = 127$, which simplifies to $12x-5 = 127$.

Step5: Use addition property of equality

Add 5 to both sides of the equation $12x-5 = 127$. The addition property of equality states that if $a=b$, then $a + c=b + c$. So, $12x-5+5=127 + 5$, which gives $12x=132$.

Step6: Use division property of equality

Divide both sides of the equation $12x=132$ by 12. The division property of equality states that if $a=b$ and $c
eq0$, then $\frac{a}{c}=\frac{b}{c}$. So, $\frac{12x}{12}=\frac{132}{12}$, and $x = 11$.

Step7: Find the measure of $\angle EFI$

Since $\angle EFI=4x^{\circ}$, substituting $x = 11$ into this expression, we get $\angle EFI=4\times11^{\circ}=44^{\circ}$.

Answer:

The proof is completed as above to show that $m\angle EFI = 44^{\circ}$.