Question

if (overrightarrow{fc}) bisects (angle bfd), (mangle bfc=(5x + 12)^{circ}), and (mangle cfd=(13x - 60)^{circ}), find the value of (mangle afe).

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{FC}$ bisects $\angle BFD$, then $m\angle BFC=m\angle CFD$. So, $5x + 12=13x-60$.

Step2: Solve the equation for $x$

First, move the $x$ - terms to one side: $12 + 60=13x-5x$. Then, $72 = 8x$. Divide both sides by 8: $x=\frac{72}{8}=9$.

Step3: Use the property of vertical - angles

$\angle AFE$ and $\angle BFC$ are vertical - angles, so $m\angle AFE=m\angle BFC$. Substitute $x = 9$ into the expression for $m\angle BFC$: $m\angle BFC=5x + 12=5\times9+12=45 + 12=57$. So, $m\angle AFE = 57^{\circ}$.

Answer:

$57^{\circ}$