Triangle Decomposition of Polygons Quick Check

Question

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Question
Use the angle sum of triangles to develop the interior angle sums of regular polygons formula using decomposition. Test the formula by finding the sum of the interior angles of a square.(1 point)
Responses

The sum of the angles of a square is 1,080°.
The sum of the angles of a square is 1,080 degrees .

The sum of the angles of a square is 182°.
The sum of the angles of a square is 182 degrees .

The sum of the angles of a square is 360°.
The sum of the angles of a square is 360 degrees .

The sum of the angles of a square is 1,440°.

Answer 1

To find the sum of the interior angles of a square (and generally for any regular polygon), we can use the formula derived from the concept of triangle decomposition.

1. Understanding the Formula: The sum of the interior angles \( S \) of a polygon with \( n \) sides can be determined using the formula: \[ S = (n - 2) \times 180° \] This formula arises because a polygon can be divided into \( n - 2 \) triangles, each having an interior angle sum of 180°.

2. Applying to a Square: A square is a regular polygon that has 4 sides (\( n = 4 \)). Using the formula: \[ S = (4 - 2) \times 180° = 2 \times 180° = 360° \]

Therefore, the sum of the angles of a square is 360°.

The correct response is:

• The sum of the angles of a square is 360°.