Question

in the figure, $overrightarrow{ca}$ and $overrightarrow{ce}$ are opposite rays, $overrightarrow{ch}$ bisects $angle gcd$, and $overrightarrow{gc}$ bisects $angle bgd$. if $mangle bgc=(6x - 13)^{circ}$ and $mangle cgf=(4x + 3)^{circ}$, what is $mangle bgf$?

Explanation:

Step1: Use angle - addition property

Since $\angle BGF=\angle BGC+\angle CGF$, we can set up an equation based on the given angle measures.
$(6x - 13)+(4x + 3)=\angle BGF$.
First, simplify the left - hand side of the equation by combining like terms.
$(6x+4x)+(-13 + 3)=10x-10$.

Step2: Solve for $x$

Since $\overrightarrow{GC}$ bisects $\angle BGD$, we know that $\angle BGC=\angle CGF$.
Set up the equation $6x - 13=4x + 3$.
Subtract $4x$ from both sides: $6x-4x-13=4x-4x + 3$, which simplifies to $2x-13 = 3$.
Add 13 to both sides: $2x-13 + 13=3 + 13$, so $2x=16$.
Divide both sides by 2: $x = 8$.

Step3: Find $\angle BGF$

Substitute $x = 8$ into the expression for $\angle BGF=10x-10$.
$\angle BGF=10\times8-10$.
$\angle BGF=80 - 10=70^{\circ}$.

Answer:

$70^{\circ}$