Question

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

Explanation:

Step1: Use angle - bisector property

Since $\overline{DE}$ 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 $m\angle GDE=m\angle EDH=(8x - 1)^{\circ}$ (or $(6x + 15)^{\circ}$), substituting $x = 8$ into $m\angle GDE$, we get $m\angle GDE=8\times8-1=63^{\circ}$. Then $m\angle GDH=2\times63^{\circ}=126^{\circ}$

Step3: Find $m\angle FDH$

We know that $\angle CDE$ is a straight - angle, so $m\angle CDE = 180^{\circ}$ and $m\angle CDF = 43^{\circ}$. Then $m\angle FDE=180^{\circ}-43^{\circ}=137^{\circ}$. Also, $m\angle FDH=m\angle FDE - m\angle EDH$. Substituting $x = 8$ into $m\angle EDH=6x + 15$, we have $m\angle EDH=6\times8+15=63^{\circ}$. So $m\angle FDH=137^{\circ}-63^{\circ}=74^{\circ}$

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:

$x = 8$
$m\angle GDH=126^{\circ}$
$m\angle FDH = 74^{\circ}$
$m\angle FDE=137^{\circ}$