Question

1. ∠fji is a straight angle. if (overrightarrow{jg}) bisects ∠fjh, find (x) and (mangle ijg). (figure not drawn to scale)

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{JG}$ bisects $\angle FJH$, then $m\angle FJG=m\angle GJH$. So, $13x - 16=8x - 26$.

Step2: Solve the equation for $x$

Subtract $8x$ from both sides: $(13x - 8x)-16=(8x - 8x)-26$, which simplifies to $5x-16=-26$. Then add 16 to both sides: $5x-16 + 16=-26 + 16$, giving $5x=-10$. Divide both sides by 5: $x=\frac{-10}{5}=-2$.

Step3: Find $m\angle IJG$

Since $\angle FJI$ is a straight - angle ($m\angle FJI = 180^{\circ}$), and $m\angle FJG=m\angle GJH$, then $m\angle IJG=m\angle GJH$. Substitute $x = - 2$ into the expression for $m\angle GJH$: $m\angle GJH=8x-26=8\times(-2)-26=-16 - 26=-42$. But angles cannot be negative in this context. We made a mistake above. The correct equation should be based on the fact that $\angle FJG+\angle GJH=\angle FJI = 180^{\circ}$. So, $(13x - 16)+(8x - 26)=180$.

Step4: Combine like terms and solve for $x$

Combine like terms: $13x+8x-16 - 26 = 180$, which gives $21x-42 = 180$. Add 42 to both sides: $21x-42 + 42=180 + 42$, so $21x=222$. Divide both sides by 21: $x=\frac{222}{21}=\frac{74}{7}\approx10.57$.

Step5: Find $m\angle IJG$

Substitute $x=\frac{74}{7}$ into the expression for $m\angle IJG$ (using $m\angle IJG = 8x-26$). $m\angle IJG=8\times\frac{74}{7}-26=\frac{592}{7}-\frac{182}{7}=\frac{592 - 182}{7}=\frac{410}{7}\approx58.57^{\circ}$.

Answer:

$x = \frac{74}{7}$, $m\angle IJG=\frac{410}{7}^{\circ}$