This geometry worksheet teaches how to calculate missing angles in irregular polygons using the sum of interior angles formula, featuring a solved example and three practice exercises.

Here are the step-by-step solutions for each polygon.

1) Pentagon (5 sides)

Step 1: Find the sum of interior angles.
Formula: $(n - 2) \times 180^\circ$
Since it has 5 sides: $(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$.

Step 2: Set up the equation.
Add all the angles together and set them equal to $540^\circ$:
$x + 105 + (x - 6) + (x + 12) + 75 = 540$

Step 3: Solve for $x$.
Combine the $x$'s: $x + x + x = 3x$
Combine the numbers: $105 - 6 + 12 + 75 = 186$
Equation: $3x + 186 = 540$
Subtract 186 from both sides: $3x = 354$
Divide by 3: $x = 118$

Step 4: Find the specific angles.
* $\angle P = x = 118^\circ$
* $\angle R = x - 6 = 118 - 6 = 112^\circ$
* $\angle S = x + 12 = 118 + 12 = 130^\circ$

***

2) Heptagon (7 sides)

Step 1: Find the sum of interior angles.
Formula: $(n - 2) \times 180^\circ$
Since it has 7 sides: $(7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$.

Step 2: Set up the equation.
Add all the angles together and set them equal to $900^\circ$:
$(x + 18) + x + 138 + (x + 10) + 130 + 130 + 132 = 900$

Step 3: Solve for $x$.
Combine the $x$'s: $x + x + x = 3x$
Combine the numbers: $18 + 138 + 10 + 130 + 130 + 132 = 558$
Equation: $3x + 558 = 900$
Subtract 558 from both sides: $3x = 342$
Divide by 3: $x = 114$

Step 4: Find the specific angles.
* $\angle A = x + 18 = 114 + 18 = 132^\circ$
* $\angle B = x = 114^\circ$
* $\angle D = x + 10 = 114 + 10 = 124^\circ$

***

3) Hexagon (6 sides)

Step 1: Find the sum of interior angles.
Formula: $(n - 2) \times 180^\circ$
Since it has 6 sides: $(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$.

Step 2: Set up the equation.
Note that angle H has a square symbol, which means it is $90^\circ$.
Add all angles and set equal to $720^\circ$:
$(x - 27) + 90 + (x + 30) + 112 + 125 + (x + 15) = 720$

Step 3: Solve for $x$.
Combine the $x$'s: $x + x + x = 3x$
Combine the numbers: $-27 + 90 + 30 + 112 + 125 + 15 = 345$
Equation: $3x + 345 = 720$
Subtract 345 from both sides: $3x = 375$
Divide by 3: $x = 125$

Step 4: Find the specific angles.
* $\angle G = x - 27 = 125 - 27 = 98^\circ$
* $\angle I = x + 30 = 125 + 30 = 155^\circ$
* $\angle L = x + 15 = 125 + 15 = 140^\circ$

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Final Answer:

1)
Sum of the interior angles = 540°
$x =$ 118
$\angle P =$ 118°
$\angle R =$ 112°
$\angle S =$ 130°

2)
Sum of the interior angles = 900°
$x =$ 114
$\angle A =$ 132°
$\angle B =$ 114°
$\angle D =$ 124°

3)
Sum of the interior angles = 720°
$x =$ 125
$\angle G =$ 98°
$\angle I =$ 155°
$\angle L =$ 140°