Question

write a two - column proof.
given (mangle gfi = 141^{circ})
prove (mangle efi = 48^{circ})
statements reasons

1. (mangle gfi = 141^{circ}) 1. given
2. (mangle gfe+mangle efi=mangle gfi) 2. angle addition postulate

addition property of equality
definition of angle bisector
video textbook get more help -

Explanation:

Step1: Apply angle - addition postulate

We know that $\angle GFI=\angle GFE+\angle EFI$. Given $\angle GFI = 141^{\circ}$, and $\angle GFE=(8x - 3)^{\circ}$, $\angle EFI = 4x^{\circ}$. So, $(8x - 3)+4x=141$.

Step2: Combine like - terms

Combining the $x$ terms on the left - hand side, we get $8x+4x-3 = 141$, which simplifies to $12x-3 = 141$.

Step3: Add 3 to both sides

Using the addition property of equality, $12x-3 + 3=141 + 3$, so $12x=144$.

Step4: Solve for x

Dividing both sides by 12, we have $\frac{12x}{12}=\frac{144}{12}$, so $x = 12$.

Step5: Find the measure of $\angle EFI$

Since $\angle EFI = 4x^{\circ}$, substituting $x = 12$ into the expression, we get $\angle EFI=4\times12^{\circ}=48^{\circ}$.

Answer:

The reason for statement 2 is Angle Addition Postulate. And we have proven that $m\angle EFI = 48^{\circ}$.