Question

if ∠cde is a straight angle, (overrightarrow{de}) bisects ∠gdh, (mangle gde=(8x - 1)^{circ}), (mangle edh=(6x + 15)^{circ}), and (mangle cdf = 43^{circ}), find each measure.
(x=) type your answer...
(mangle gdh=) type your answer...(^{circ})
(mangle fdh=) type your answer...(^{circ})
(mangle fde=) type your answer...(^{circ})

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{DE}$ bisects $\angle GDH$, then $m\angle GDE=m\angle EDH$. So, we set up the equation $8x - 1=6x + 15$.
$8x-6x=15 + 1$
$2x=16$
$x = 8$

Step2: Find $m\angle GDH$

Since $m\angle GDH=m\angle GDE + m\angle EDH$ and $m\angle GDE=m\angle EDH$, when $x = 8$, $m\angle GDE=8x-1=8\times8 - 1=63^{\circ}$, $m\angle EDH=6x + 15=6\times8+15=63^{\circ}$. Then $m\angle GDH=63^{\circ}+63^{\circ}=126^{\circ}$.

Step3: Find $m\angle FDH$

We know that $\angle CDE$ is a straight - angle, so $m\angle CDE = 180^{\circ}$. Given $m\angle CDF = 43^{\circ}$, then $m\angle FDE=180^{\circ}-43^{\circ}=137^{\circ}$. Also, $m\angle FDH=m\angle FDE - m\angle EDH$. Substituting $m\angle FDE = 137^{\circ}$ and $m\angle EDH = 63^{\circ}$, we get $m\angle FDH=137^{\circ}-63^{\circ}=74^{\circ}$.

Step4: Find $m\angle FDE$

As $\angle CDE$ is a straight - angle ($m\angle CDE = 180^{\circ}$) and $m\angle CDF = 43^{\circ}$, then $m\angle FDE=180^{\circ}-43^{\circ}=137^{\circ}$.

Answer:

$x = 8$
$m\angle GDH = 126$
$m\angle FDH = 74$
$m\angle FDE = 137$