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Problem Overview:

We are tasked with finding the missing angles in three irregular polygons. To solve this, we will use the formula for the sum of the interior angles of a polygon:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon. Then, we will use the given angle expressions to solve for the unknown variable \( x \) and determine the missing angles.

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Step-by-Step Solution:

#### 1. Polygon 1:
The polygon has 5 sides (a pentagon).

##### Step 1: Calculate the sum of the interior angles.
\[
\text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

##### Step 2: Write the equation for the sum of the interior angles.
The angles are given as:
- \( 105^\circ \)
- \( x \)
- \( 75^\circ \)
- \( x + 12^\circ \)
- \( x - 6^\circ \)

The sum of these angles is:
\[
105^\circ + x + 75^\circ + (x + 12^\circ) + (x - 6^\circ) = 540^\circ
\]

##### Step 3: Simplify the equation.
Combine like terms:
\[
105^\circ + 75^\circ + 12^\circ - 6^\circ + x + x + x = 540^\circ
\]
\[
186^\circ + 3x = 540^\circ
\]

##### Step 4: Solve for \( x \).
\[
3x = 540^\circ - 186^\circ
\]
\[
3x = 354^\circ
\]
\[
x = \frac{354^\circ}{3} = 118^\circ
\]

##### Step 5: Find the missing angles.
- \( \angle P = x = 118^\circ \)
- \( \angle R = x + 12^\circ = 118^\circ + 12^\circ = 130^\circ \)
- \( \angle S = x - 6^\circ = 118^\circ - 6^\circ = 112^\circ \)

##### Final Answer for Polygon 1:
\[
\boxed{540^\circ, x = 118^\circ, \angle P = 118^\circ, \angle R = 130^\circ, \angle S = 112^\circ}
\]

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#### 2. Polygon 2:
The polygon has 6 sides (a hexagon).

##### Step 1: Calculate the sum of the interior angles.
\[
\text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]

##### Step 2: Write the equation for the sum of the interior angles.
The angles are given as:
- \( x + 18^\circ \)
- \( 132^\circ \)
- \( 130^\circ \)
- \( 130^\circ \)
- \( 138^\circ \)
- \( x + 10^\circ \)

The sum of these angles is:
\[
(x + 18^\circ) + 132^\circ + 130^\circ + 130^\circ + 138^\circ + (x + 10^\circ) = 720^\circ
\]

##### Step 3: Simplify the equation.
Combine like terms:
\[
x + 18^\circ + 132^\circ + 130^\circ + 130^\circ + 138^\circ + x + 10^\circ = 720^\circ
\]
\[
2x + (18^\circ + 132^\circ + 130^\circ + 130^\circ + 138^\circ + 10^\circ) = 720^\circ
\]
\[
2x + 558^\circ = 720^\circ
\]

##### Step 4: Solve for \( x \).
\[
2x = 720^\circ - 558^\circ
\]
\[
2x = 162^\circ
\]
\[
x = \frac{162^\circ}{2} = 81^\circ
\]

##### Step 5: Find the missing angles.
- \( \angle A = x + 18^\circ = 81^\circ + 18^\circ = 99^\circ \)
- \( \angle B = x + 10^\circ = 81^\circ + 10^\circ = 91^\circ \)
- \( \angle D = 130^\circ \) (already given)

##### Final Answer for Polygon 2:
\[
\boxed{720^\circ, x = 81^\circ, \angle A = 99^\circ, \angle B = 91^\circ, \angle D = 130^\circ}
\]

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#### 3. Polygon 3:
The polygon has 7 sides (a heptagon).

##### Step 1: Calculate the sum of the interior angles.
\[
\text{Sum of interior angles} = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
\]

##### Step 2: Write the equation for the sum of the interior angles.
The angles are given as:
- \( x + 27^\circ \)
- \( 90^\circ \) (right angle)
- \( x + 30^\circ \)
- \( 125^\circ \)
- \( x + 15^\circ \)
- \( 112^\circ \)
- \( x \)

The sum of these angles is:
\[
(x + 27^\circ) + 90^\circ + (x + 30^\circ) + 125^\circ + (x + 15^\circ) + 112^\circ + x = 900^\circ
\]

##### Step 3: Simplify the equation.
Combine like terms:
\[
x + 27^\circ + 90^\circ + x + 30^\circ + 125^\circ + x + 15^\circ + 112^\circ + x = 900^\circ
\]
\[
4x + (27^\circ + 90^\circ + 30^\circ + 125^\circ + 15^\circ + 112^\circ) = 900^\circ
\]
\[
4x + 399^\circ = 900^\circ
\]

##### Step 4: Solve for \( x \).
\[
4x = 900^\circ - 399^\circ
\]
\[
4x = 501^\circ
\]
\[
x = \frac{501^\circ}{4} = 125.25^\circ
\]

##### Step 5: Find the missing angles.
- \( \angle G = x + 27^\circ = 125.25^\circ + 27^\circ = 152.25^\circ \)
- \( \angle I = x + 30^\circ = 125.25^\circ + 30^\circ = 155.25^\circ \)
- \( \angle L = x + 15^\circ = 125.25^\circ + 15^\circ = 140.25^\circ \)
- \( \angle 1 = x = 125.25^\circ \)

##### Final Answer for Polygon 3:
\[
\boxed{900^\circ, x = 125.25^\circ, \angle G = 152.25^\circ, \angle I = 155.25^\circ, \angle L = 140.25^\circ, \angle 1 = 125.25^\circ}
\]

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Final Answers:

1. \(\boxed{540^\circ, x = 118^\circ, \angle P = 118^\circ, \angle R = 130^\circ, \angle S = 112^\circ}\)
2. \(\boxed{720^\circ, x = 81^\circ, \angle A = 99^\circ, \angle B = 91^\circ, \angle D = 130^\circ}\)
3. \(\boxed{900^\circ, x = 125.25^\circ, \angle G = 152.25^\circ, \angle I = 155.25^\circ, \angle L = 140.25^\circ, \angle 1 = 125.25^\circ}\)