Question

refer to the figure. if (mangle bdc=(8x + 12)^{circ}) and (mangle fdb=(12x - 32)^{circ}), find (mangle fde). assume the figure is not drawn to scale.

Explanation:

Step1: Note angle - relationship

$\angle BDC$ and $\angle FDB$ are supplementary, so $\angle BDC+\angle FDB = 180^{\circ}$.

Step2: Set up the equation

$(8x + 12)+(12x-32)=180$.

Step3: Combine like - terms

$8x+12x+12 - 32=180$, which simplifies to $20x-20 = 180$.

Step4: Solve for x

Add 20 to both sides: $20x=180 + 20=200$. Then divide both sides by 20, so $x = 10$.

Step5: Find $\angle FDB$

Substitute $x = 10$ into the expression for $\angle FDB$: $m\angle FDB=(12x - 32)^{\circ}=(12\times10-32)^{\circ}=(120 - 32)^{\circ}=88^{\circ}$.

Step6: Note $\angle FDE$ and $\angle FDB$ relationship

$\angle FDE$ and $\angle FDB$ are vertical angles, so $m\angle FDE=m\angle FDB$.

Answer:

$88^{\circ}$