The included angle between two successive sides of a regular eight-sided polygon is a) \(150^{\circ}\) b) \(135^{\circ}\) c) \(120^{\circ}\) d) \(75^{\circ}\)

Understanding the sum of interior angles formula is crucial when tackling geometry problems involving polygons. This formula, which is \(S = (n - 2) \times 180^{\text{o}}\), allows us to easily calculate the sum (\(S\)) of all the interior angles of any polygon by simply knowing the number of sides (\(n\)).

In the context of a regular polygon, which is a polygon with all sides and angles equal, this formula becomes especially powerful as it helps us to determine the measure of each individual angle. For example, with an eight-sided polygon, known as an octagon, we can substitute \(n = 8\) into the formula and get \(S = (8 - 2) \times 180^{\text{o}} = 6 \times 180^{\text{o}} = 1080^{\text{o}}\).

Remember, this formula derives from the concept that any polygon can be divided into triangles, and since each triangle's interior angles sum to \(180^{\text{o}}\), multiplying the number of triangles (\(n - 2\)) by \(180^{\text{o}}\) gives us the total interior angles sum for the polygon.

Polygons are a foundational element in the study of geometry. In essence, a polygon is a closed plane figure with at least three straight sides and angles. There are various types of polygons, classified by the number of sides they have—triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and so on, up to n-sided polygons. The properties and classification further extend to whether a polygon is 'regular' or 'irregular'.

A regular polygon has all sides and angles equal, making calculations and problem-solving simpler due to this uniformity. Irregular polygons, on the other hand, have sides and angles of different lengths and measures respectively, necessitating more complex methods to arrive at solutions.

Geometry problem solving often involves identifying these types of properties as they can greatly simplify or indicate the approach needed for a given problem. For instance, knowing that an eight-sided polygon is regular immediately tells us that all angles are equal, and thus, we can use the sum of interior angles formula efficiently.

Approaching problems in geometry, particularly those involving polygon interior angles, means strategically applying formulas and properties that relate to these shapes. Geometry problem solving involves a step-by-step reasoning process where recognizing patterns and applying known formulas are key to finding solutions.

Using the example of the regular eight-sided polygon from the exercise, once we've calculated the sum of interior angles using the appropriate formula, the next logical step is to divide this sum by the number of sides. This process highlights a systematic approach where each step builds on the previous one towards reaching the final answer.

Enhancing Problem-Solving Skills

Practice is vital. Tackling a variety of problems helps in understanding the diverse applications of geometry concepts. Always verify your final answer against the situation being modeled; in real-world applications, answers need to make practical sense. Above all, developing a keen sense of observation to notice properties of shapes in the problems you solve can lead to more intuitive and quicker solutions.