Question

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

Explanation:

Step1: Use angle - bisector property

Since $\overline{DE}$ 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 simplifies to $2x-1 = 15$. 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 $m\angle{GDE}$ or $m\angle{EDH}$

Substitute $x = 8$ into the expression for $m\angle{GDE}$: $m\angle{GDE}=8x-1=8\times8 - 1=64 - 1=63^{\circ}$.

Step4: Calculate $m\angle{GDH}$

Since $m\angle{GDH}=m\angle{GDE}+m\angle{EDH}$ and $m\angle{GDE}=m\angle{EDH}$, then $m\angle{GDH}=2m\angle{GDE}$. So $m\angle{GDH}=2\times63^{\circ}=126^{\circ}$.

Answer:

$126^{\circ}$