For a regular pentagon (a five sided polygon with all sides and angles equal), how many distinct lines of symmetry does the figure have?

Introduction / Context:
Lines of symmetry play an important role in geometry and non verbal reasoning. A line of symmetry is a line that divides a shape into two mirror image halves. Regular polygons, where all sides and angles are equal, have a predictable number of lines of symmetry. Here we consider a regular pentagon and determine how many symmetry lines it has.

Given Data / Assumptions:

• The figure is a regular pentagon, meaning five equal sides and five equal interior angles.
• We are asked about lines of symmetry, not rotational symmetry.
• A line of symmetry must produce two identical halves when the figure is folded along that line.
• The pentagon is not irregular or skewed; it is perfectly regular.

Concept / Approach:
For regular polygons, the number of lines of symmetry equals the number of sides. Each line of symmetry either passes through one vertex and the midpoint of the opposite side or, in some polygons, through midpoints of opposite sides. For a regular pentagon there are five vertices and five sides, and each vertex can be matched symmetrically with the opposite side.

Step-by-Step Solution:
Step 1: Recall that a pentagon has 5 sides by definition. Step 2: For a regular polygon with n sides, there are n lines of symmetry. Step 3: In a regular pentagon, each line of symmetry passes through one vertex and the midpoint of the opposite side. Step 4: Because the pentagon has 5 vertices, we can draw exactly 5 such lines. Step 5: No other lines will create mirror image halves, because any other line will intersect sides or angles in an asymmetric way. Step 6: Therefore the number of lines of symmetry in a regular pentagon is 5.

Verification / Alternative check:
You can draw a regular pentagon and construct these lines using a ruler. Draw a straight line from each vertex to the midpoint of the opposite side. Folding the paper along any of these lines shows that one half falls exactly over the other, confirming that each line is a line of symmetry. Counting all such distinct lines gives 5, matching the theoretical rule of n symmetry lines for an n sided regular polygon.

Why Other Options Are Wrong:

• 4: This would be correct for some other figures but not for a regular pentagon. It undercounts by one.
• 6: This overcounts and would suggest more symmetry than the shape actually has.
• 2: A rectangle has 2 lines of symmetry (unless it is a square), but this does not apply here.
• 0: A completely irregular pentagon can have no line of symmetry, but the question explicitly refers to a regular pentagon.

Common Pitfalls:
Learners sometimes confuse the number of sides with possible diagonals or think that odd sided shapes cannot have symmetry lines. Others may try to draw lines between vertices only, missing the vertex to midpoint lines. Remembering the general rule that a regular n sided polygon has n lines of symmetry is very helpful in such questions.

Final Answer:
A regular pentagon has 5 lines of symmetry.