Question

find the value of each variable
10.
11

Explanation:

Step1: Solve Problem 10 (polygon exterior angles)

The sum of exterior angles of any polygon is $360^\circ$.
$$2x + 3x + 4x + x = 360$$
$$10x = 360$$
$$x = 36$$

Step2: Solve Problem 11 (heptagon interior angles)

First, calculate the total interior angles of a 7-sided polygon using $(n-2)\times180^\circ$ where $n=7$:
$$(7-2)\times180 = 900$$
Sum the given angles and set equal to 900:
$$(n-17)+(n+30)+142+(n+61)+128+139+n = 900$$
Combine like terms:
$$5n + (-17+30+142+61+128+139) = 900$$
$$5n + 483 = 900$$
$$5n = 417$$
$$n = 83.4$$

Step3: Solve Problem 12 (octagon interior angles)

Calculate total interior angles for 8-sided polygon ($n=8$):
$$(8-2)\times180 = 1080$$
Let the unknown angle be $y$, sum given angles and set equal to 1080:
$$130+140+150+135+125+105+145+y = 1080$$
Sum the known angles:
$$130+140+150+135+125+105+145 = 930$$
Solve for $y$:
$$y = 1080 - 930 = 150$$

Answer:

Problem 10: $x = 36$
Problem 11: $n = 83.4$
Problem 12: The unknown angle is $150^\circ$