Question

1. ∠fij is a straight angle. if (overrightarrow{ij}) bisects ∠fij, find x and m∠1 and m∠g. (figure not drawn to scale)

Explanation:

Step1: Recall angle - bisector property

Since $\overrightarrow{JG}$ bisects $\angle FJI$, then $\angle FJG=\angle GJI$. So, $12x - 16=8x - 26$.

Step2: Solve the equation for $x$

Subtract $8x$ from both sides: $(12x - 8x)-16=(8x - 8x)-26$, which simplifies to $4x-16=-26$.
Then add 16 to both sides: $4x-16 + 16=-26 + 16$, so $4x=-10$.
Divide both sides by 4: $x=\frac{-10}{4}=-\frac{5}{2}$.

Step3: Find $m\angle FJG$

Substitute $x =-\frac{5}{2}$ into the expression for $\angle FJG$: $m\angle FJG=12x - 16$.
$m\angle FJG=12\times(-\frac{5}{2})-16=-30 - 16=-46$. But angles cannot be negative in this context, we made a mistake above. Let's correct it.
Since $\angle FJI$ is a straight - angle, $\angle FJG+\angle GJI = 180^{\circ}$ (because $\overrightarrow{JG}$ bisects $\angle FJI$). So, $(12x - 16)+(8x - 26)=180$.

Step4: Combine like terms

$12x+8x-16 - 26 = 180$, which gives $20x-42 = 180$.

Step5: Solve for $x$

Add 42 to both sides: $20x-42 + 42=180 + 42$, so $20x=222$.
Divide both sides by 20: $x=\frac{222}{20}=\frac{111}{10}=11.1$.

Step6: Find $m\angle FJG$

Substitute $x = 11.1$ into the expression for $\angle FJG$: $m\angle FJG=12x - 16$.
$m\angle FJG=12\times11.1-16=133.2-16 = 117.2^{\circ}$.

Step7: Find $m\angle GJI$

Substitute $x = 11.1$ into the expression for $\angle GJI$: $m\angle GJI=8x - 26$.
$m\angle GJI=8\times11.1-26=88.8-26 = 62.8^{\circ}$.

Answer:

$x = 11.1$, $m\angle FJG=117.2^{\circ}$, $m\angle GJI = 62.8^{\circ}$