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.

x =
m∠gdh=
m∠fdh=
m∠fde =

Explanation:

Step1: Use angle - bisector property

Since $DE$ bisects $\angle GDH$, we have $m\angle GDE=m\angle EDH$. So, $8x + 1=6x+15$.
$$8x+1 = 6x + 15$$

Step2: Solve the equation for $x$

Subtract $6x$ from both sides: $8x-6x+1=6x - 6x+15$, which gives $2x+1 = 15$. Then subtract 1 from both sides: $2x+1 - 1=15 - 1$, so $2x=14$. Divide both sides by 2: $x = 7$.
$$2x=14\Rightarrow x = 7$$

Step3: Find $m\angle GDH$

Since $m\angle GDH=m\angle GDE + m\angle EDH$ and $m\angle GDE=m\angle EDH=8x + 1$ (when $x = 7$), $m\angle GDE=8\times7+1=57^{\circ}$, so $m\angle GDH=2\times57^{\circ}=114^{\circ}$.
$$m\angle GDH = 2(8x + 1)=2(8\times7+1)=114^{\circ}$$

Step4: Find $m\angle FDH$

Since $\angle CDE$ is a straight - angle ($180^{\circ}$) and $m\angle CDF = 43^{\circ}$, then $m\angle FDE=180^{\circ}-43^{\circ}=137^{\circ}$. Also, $m\angle FDH=m\angle FDE - m\angle EDH$. We know $m\angle EDH=6x + 1=6\times7+1 = 43^{\circ}$, so $m\angle FDH=137^{\circ}-43^{\circ}=94^{\circ}$.
$$m\angle FDH=180^{\circ}-43^{\circ}-(6x + 1)=180 - 43-(6\times7 + 1)=94^{\circ}$$

Step5: Find $m\angle FDE$

Since $\angle CDE$ is a straight - angle, $m\angle FDE=180^{\circ}-m\angle CDF$. So $m\angle FDE=180^{\circ}-43^{\circ}=137^{\circ}$.
$$m\angle FDE=180 - 43=137^{\circ}$$

Answer:

$x = 7$
$m\angle GDH=114^{\circ}$
$m\angle FDH = 94^{\circ}$
$m\angle FDE=137^{\circ}$