Question

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

Explanation:

Step1: Use angle - bisector property

Since $\overline{DE}$ bisects $\angle{GDH}$, then $m\angle{GDE}=m\angle{EDH}$. So, $8x - 1=6x + 15$.

Step2: Solve for $x$

$8x-6x=15 + 1$, $2x=16$, $x = 8$.

Step3: Find $m\angle{GDH}$

$m\angle{GDH}=m\angle{GDE}+m\angle{EDH}$, $m\angle{GDE}=8x - 1=8\times8 - 1=63^{\circ}$, $m\angle{GDH}=2\times63^{\circ}=126^{\circ}$.

Step4: Find $m\angle{FDH}$

$m\angle{CDF}+m\angle{FDH}=180^{\circ}$ (linear - pair), $m\angle{FDH}=180^{\circ}-43^{\circ}=137^{\circ}$.

Step5: Find $m\angle{FDE}$

$m\angle{FDE}=m\angle{CDF}+m\angle{CDE}$, $m\angle{CDE}=180^{\circ}$, $m\angle{FDE}=43^{\circ}+180^{\circ}=223^{\circ}$ (or considering the non - reflex angle, $m\angle{FDE}=180^{\circ}-43^{\circ}=137^{\circ}$).

Answer:

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