Question

1. if ∠cde is a straight angle, 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 $DE$ bisects $\angle GDH$, we have $m\angle GDE=m\angle EDH$. So, $8x - 1=6x + 15$.

Step2: Solve the equation for $x$

Subtract $6x$ from both sides: $8x-6x - 1=6x-6x + 15$, which gives $2x-1 = 15$. Then add 1 to both sides: $2x-1 + 1=15 + 1$, so $2x=16$. Divide both sides by 2: $x=\frac{16}{2}=8$.

Step3: Find $m\angle GDH$

Since $m\angle GDH=m\angle GDE + m\angle EDH$ and $m\angle GDE=m\angle EDH = 8x - 1$ (or $6x + 15$), when $x = 8$, $m\angle GDE=8\times8 - 1=63^{\circ}$, so $m\angle GDH=2\times63^{\circ}=126^{\circ}$.

Step4: 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, $m\angle FDH=m\angle FDE - m\angle EDH$. Since $m\angle EDH=6x + 15$ and $x = 8$, $m\angle EDH=6\times8 + 15=63^{\circ}$. So $m\angle FDH=137^{\circ}-63^{\circ}=74^{\circ}$.

Step5: Find $m\angle FDE$

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

Answer:

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