Consider the following statements about similar triangles and polygons. Which one of the following statements is true?

Introduction / Context:
This is a conceptual question from geometry about the definition and properties of similar triangles and polygons. It asks you to identify which statement correctly describes when triangles are similar and to carefully distinguish similarity from congruence and equal area.

Given Data / Assumptions:

• We deal with triangles and polygons in Euclidean geometry.
• Four candidate statements about similarity are given.
• We must select the single correct statement.

Concept / Approach:
Two figures are similar if they have the same shape. For triangles, a key criterion is that corresponding sides are in proportion and corresponding angles are equal. In fact, for triangles, proportionality of corresponding sides alone is sufficient to guarantee similarity because angle equality follows automatically.

Step-by-Step Solution:
Step 1: Review option (a): Similar triangles have equal shape but can be of different size, so they are not always congruent. Thus (a) is false.Step 2: Option (b): Similar triangles can be scaled versions of each other, so their areas are generally not equal. This is false.Step 3: Option (c): For triangles, if the ratios of corresponding sides are equal, then the triangles are similar. This is a standard similarity criterion, so (c) is true.Step 4: Option (d): For general polygons, proportional sides alone do not guarantee similarity; corresponding angles must also be equal. Hence this statement is incomplete and false.

Verification / Alternative check:
You can recall standard triangle similarity tests such as side side side similarity, angle side angle similarity and angle angle angle similarity. The side side side similarity test directly states that proportionality of corresponding sides implies similarity, supporting option (c).

Why Other Options Are Wrong:
Options (a) and (e) incorrectly equate similarity with congruence. Option (b) incorrectly states equal areas, which is not required. Option (d) ignores the angle condition for polygons with more than three sides.

Common Pitfalls:
Many learners confuse similarity and congruence, thinking that same shape always implies same size. Others wrongly assume equal area means similarity. Remember that similar figures can be scaled versions of each other, preserving shape but not necessarily size or area.

Final Answer:
The correct statement is Two triangles are similar if their corresponding sides are proportional.