Question

(9) find ( mangle f ) the figure is a parallelogram labeled with vertices f, g, h, e. angle at g is ( (2y - 47)^circ ), angle at e is ( (y + 35)^circ )

Explanation:

Step1: Use parallelogram consecutive angles property

In a parallelogram, consecutive angles are supplementary, so:
$$(2y-47)^\circ + (y+35)^\circ = 180^\circ$$

Step2: Simplify and solve for $y$

Combine like terms and isolate $y$:
$$3y - 12 = 180$$
$$3y = 192$$
$$y = 64$$

Step3: Find $\angle G$ first

Substitute $y=64$ into $\angle G$ expression:
$$m\angle G = (2(64)-47)^\circ = (128-47)^\circ = 81^\circ$$

Step4: Use parallelogram opposite angles property

In a parallelogram, opposite angles are equal, so $\angle F = \angle H$, and $\angle E = \angle G$. Also, $\angle F$ is supplementary to $\angle E$:
$$m\angle F = 180^\circ - m\angle E$$
First calculate $m\angle E$:
$$m\angle E = (64+35)^\circ = 99^\circ$$
$$m\angle F = 180^\circ - 99^\circ = 81^\circ$$
(Alternatively, since $\angle F$ is opposite $\angle H$, and $\angle H$ is supplementary to $\angle G$, we get the same result)

Answer:

$81^\circ$