Question

1. find x.

the figure is a nonagon (9 - sided polygon) with the following angle expressions: (5x - 30), (x), (2x), (8x - 120), (x + 30), (4x), (2x - 18), (x - 12), (2x + 52), (3x + 2) (wait, actually, counting the sides: lets list the given angle labels on the nonagon: (3x + 2), (5x - 30), (x), (2x), (8x - 120), (x + 30), (4x), (2x - 18), (x - 12), (2x + 52) – wait, maybe i miscounted, but the key is its a polygon (nonagon, 9 sides) with angle measures as algebraic expressions, and we need to find (x) by using the sum of interior angles formula for a polygon.
wait, the ocr text from the image: the problem is to find (x) for a nonagon (9 - sided) with angles labeled as (3x + 2), (5x - 30), (x), (2x), (8x - 120), (x + 30), (4x), (2x - 18), (x - 12), (2x + 52)? wait, no, looking at the image again: the nonagon has the following angle expressions (lets count the number of sides: the labels are: (3x + 2), (5x - 30), (x), (2x), (8x - 120), (x + 30), (4x), (2x - 18), (x - 12), (2x + 52) – wait, maybe its a 9 - sided, so 9 angles. lets list them:

1. (3x + 2)
2. (5x - 30)
3. (x)
4. (2x)
5. (8x - 120)
6. (x + 30)
7. (4x)
8. (2x - 18)
9. (x - 12)
10. wait, no, maybe i made a mistake. wait the image shows a nonagon (9 sides), so 9 angles. lets check the original image again: the labels are: (3x + 2), (2x + 52), (x - 12), (2x - 18), (4x), (x + 30), (8x - 120), (2x), (x), (5x - 30) – yes, thats 9 angles. so the 9 angle expressions are: (3x + 2), (2x + 52), (x - 12), (2x - 18), (4x), (x + 30), (8x - 120), (2x), (x), (5x - 30)? wait, no, the users image: lets parse the ocr correctly. the problem is: \7. find x.\ and the figure is a nonagon (9 - sided) with the following angle measures (as algebraic expressions): (3x + 2), (2x + 52), (x - 12), (2x - 18), (4x), (x + 30), (8x - 120), (2x), (x), (5x - 30)? wait, maybe the correct count is 9 angles. lets list them as per the image:

• left side: (3x + 2), then (5x - 30), then (x), then (2x), then (8x - 120), then (x + 30), then (4x), then (2x - 18), then (x - 12), then (2x + 52) – wait, thats 10? no, maybe its a nonagon (9 sides), so perhaps a typo, but the key is that its a polygon with (n) sides, and we need to use the sum of interior angles formula: ((n - 2) \times 180^circ), then set the sum of the given angle expressions equal to that, and solve for (x).

so the ocr text is: \7. find x.\ followed by the nonagon with angle expressions: (3x + 2), (5x - 30), (x), (2x), (8x - 120), (x + 30), (4x), (2x - 18), (x - 12), (2x + 52) (assuming 9 sides, maybe one is a repeat or mislabel, but the problem is to find (x) using the sum of interior angles of a polygon).

Explanation:

Step1: Calculate total interior angle

First, identify the polygon is a decagon (10 sides). The formula for total interior angles is $(n-2)\times180^\circ$ where $n=10$.
$$(10-2)\times180^\circ = 8\times180^\circ = 1440^\circ$$

Step2: Sum all given angle expressions

Add up all the angle terms and combine like terms.

$$\begin{align*} &(5x-30)+x+2x+(8x-120)+(x+30)+4x+(2x-18)+(x-12)+(2x+52)+(3x+2)\\ =&5x+x+2x+8x+x+4x+2x+x+2x+3x + (-30-120+30-18-12+52+2)\\ =&31x - 96 \end{align*}$$

Step3: Set equal to total angle, solve for x

Equate the summed expression to 1440 and isolate x.

$$\begin{align*} 31x - 96 &= 1440\\ 31x &= 1440 + 96\\ 31x &= 1536\\ x &= \frac{1536}{31} \end{align*}$$

Answer:

$x=\frac{1536}{31}$ or approximately $49.55$