Question

write a two - column proof. given: m∠gfi = 117°. prove: m∠efi = 30°. statements reasons 1. m∠gfi = 117° 1. given 2. m∠gfe + m∠efi = m∠gfi 2. 3. m∠gfe + m∠efi = ° 3. substitution property. (steps 1, 2) 4. m∠gfe=(9x - 3)°, m∠efi = 3x° 4. given 5. 9x - 3+3x = 117 5. 6. 6. combine like terms. 7. x = 7. addition property of equality 8. x = 8. division property of equality 9. m∠efi = ° 9. substitution property. (steps 4, 8)

Explanation:

Step1: Identify angle - addition postulate

The sum of adjacent angles $\angle GFE$ and $\angle EFI$ equals $\angle GFI$. So, the reason for statement 2 is the Angle - Addition Postulate.

Step2: Substitute values

Since $m\angle GFI = 117^{\circ}$, substituting into $m\angle GFE + m\angle EFI=m\angle GFI$ gives $m\angle GFE + m\angle EFI = 117^{\circ}$.

Step3: Substitute expressions for angles

Given $m\angle GFE=(9x - 3)^{\circ}$ and $m\angle EFI = 3x^{\circ}$, we get the equation $9x-3 + 3x=117$.

Step4: Combine like - terms

Combining the $x$ terms on the left - hand side of the equation $9x-3 + 3x=117$ gives $12x-3 = 117$.

Step5: Apply addition property of equality

Add 3 to both sides of the equation $12x-3 = 117$. So, $12x-3+3=117 + 3$, which simplifies to $12x=120$.

Step6: Apply division property of equality

Divide both sides of the equation $12x = 120$ by 12. So, $\frac{12x}{12}=\frac{120}{12}$, and $x = 10$.

Step7: Find $m\angle EFI$

Since $m\angle EFI = 3x^{\circ}$ and $x = 10$, substituting $x = 10$ into the expression for $m\angle EFI$ gives $m\angle EFI=3\times10^{\circ}=30^{\circ}$.

Answer:

1. Angle - Addition Postulate
2. 117
3. Substitution of angle expressions
4. $12x-3 = 117$
5. 120
6. 10
7. 30