Let's solve the problem for each irregular polygon step-by-step:

Polygon (a):

• This shape is a quadrilateral.
• We know that the sum of the interior angles of any quadrilateral is $360^\circ$.
• Two of the angles are right angles ($90^\circ$) and the other two angles are marked as $x$.

To find $x$, we apply the formula for the sum of interior angles of a quadrilateral: $x + x + 90^\circ + 90^\circ = 360^\circ$ $2x + 180^\circ = 360^\circ$ $2x = 180^\circ$ $x = 90^\circ$

Thus, for figure (a), the value of $x$ is $90^\circ$.

Polygon (b):

• This is a heptagon (7-sided polygon).
• The sum of the interior angles of a heptagon is calculated by the formula: $\text{Sum of interior angles} = 180^\circ \times (n-2)$ where $n$ is the number of sides. $\text{Sum of interior angles} = 180^\circ \times (7-2) = 180^\circ \times 5 = 900^\circ$

Now, let's add up the known angles: $140^\circ + 130^\circ + 130^\circ + 120^\circ + 100^\circ + 170^\circ = 790^\circ$

To find $x$, subtract the sum of the given angles from the total sum: $x = 900^\circ - 790^\circ = 110^\circ$

Thus, for figure (b), the value of $x$ is $110^\circ$.

Polygon (c):

• This is a pentagon (5-sided polygon).
• We can apply the sum of interior angles formula again: $\text{Sum of interior angles} = 180^\circ \times (5-2) = 180^\circ \times 3 = 540^\circ$

Let's add up the known angles in this pentagon: $2x + 2x + 72^\circ + 90^\circ + 90^\circ = 540^\circ$ $4x + 252^\circ = 540^\circ$ $4x = 288^\circ$ $x = 72^\circ$

Thus, for figure (c), the value of $x$ is $72^\circ$.

Summary of Results:

• (a) $x = 90^\circ$
• (b) $x = 110^\circ$
• (c) $x = 72^\circ$

Would you like a further explanation of any of the steps?

Here are five related questions for practice:

1. What is the sum of the interior angles of a hexagon?
2. How do you find the value of a missing angle in a quadrilateral if the other three angles are given?
3. What is the sum of the interior angles of an octagon?
4. How do you calculate the sum of exterior angles of any polygon?
5. What is the relationship between the number of sides of a polygon and its interior angle sum?

Tip: Always remember that the sum of the exterior angles of any polygon is always $360^\circ$, regardless of the number of sides.