Question

given ∠cfd is a right - angle, find each angle.

1. (mangle bfc = 7x - 40), (mangle cfe = 2x + 85). find (mangle efd).

Explanation:

Step1: Use angle - addition property

Since $\angle BFC+\angle CFE = 180^{\circ}$ (linear - pair of angles), we set up the equation $(7x - 40)+(2x + 85)=180$.
$$(7x - 40)+(2x + 85)=180$$

Step2: Combine like - terms

Combine the $x$ terms and the constant terms: $7x+2x-40 + 85=180$, which simplifies to $9x + 45=180$.
$$9x+45 = 180$$

Step3: Solve for $x$

Subtract 45 from both sides: $9x=180 - 45=135$. Then divide both sides by 9: $x=\frac{135}{9}=15$.
$$x = 15$$

Step4: Find $\angle CFE$

Substitute $x = 15$ into the expression for $\angle CFE$: $\angle CFE=2x + 85=2\times15+85=30 + 85 = 115^{\circ}$.
$$\angle CFE=115^{\circ}$$

Step5: Use the right - angle relationship

Since $\angle CFD = 90^{\circ}$ and $\angle CFE+\angle EFD+\angle CFD=360^{\circ}$ (a full - circle is $360^{\circ}$), and considering the angles around point $F$, we know that $\angle CFE+\angle EFD = 270^{\circ}$ (because the non - relevant part of the circle around $F$ is the right - angle $\angle CFD$). So $\angle EFD=270^{\circ}-\angle CFE$.
Substitute $\angle CFE = 115^{\circ}$ into the equation: $\angle EFD=270-115 = 155^{\circ}$.

Answer:

$155$