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A regular polygon is a geometric figure with equal-length sides and equal-measured angles. They are symmetrical shapes, meaning each angle and side is congruent to its peers. Examples of regular polygons include equilateral triangles (three sides), squares (four sides), and pentagons (five sides).

To characterize regular polygons, we look at:

• Sides: All sides of a regular polygon are equal.
• Angles: All interior angles of a regular polygon are the same.
• Rotational Symmetry: Regular polygons look the same even when rotated about their center at multiples of a specific angle.

Understanding these properties helps in creating proofs involving regular polygons, like determining possible angle measures.

Interior angles are the angles found inside a polygon. For regular polygons, these angles are all equal. Calculating the measure of these angles is crucial for solving problems involving polygons.

The formula to determine the measure of an interior angle of a regular polygon is given by \[180 - \frac{360}{n}\]where \(n\) is the number of sides in the polygon. This formula is derived from the total sum of the interior angles of a polygon, \[(n-2) \times 180,\]which is then divided by the number of angles, \(n\), since each angle is equal in a regular polygon.

For instance, in a hexagon (a polygon with 6 sides), each interior angle would be \[180 - \frac{360}{6} = 120\] degrees. This universal formula allows us to understand relationships between angles and sides in geometric figures effectively.

Mathematical reasoning is the thought process that allows us to solve problems and prove statements through logical steps. In the context of geometry, it involves setting up equations based on known formulas and then logically deducing conclusions.

Take the given problem as an example: it requires us to understand and apply the formula for the interior angle of a regular polygon. By setting the angle to 155 degrees and solving for \(n\) (number of sides), we observe:\[180 - \frac{360}{n} = 155\]

This manipulation leads us to find:\[n = \frac{360}{25} = 14.4\]

Since 14.4 is not a feasible number of sides for a polygon, we use our logical reasoning to conclude that no regular polygon has an interior angle of 155 degrees. Mathematical reasoning helps us identify and correct impossible scenarios in geometry.