Question

complete the table to show the number of triangles that each polygon can be decomposed into and the interior angle sum for each polygon.

number of sides of the polygon	number of triangles formed	interior angle sum
13	\\(\square\\)	1,980\\(^\circ\\)
14	12	\\(\square^\circ\\)
15	\\(\square\\)	\\(\square^\circ\\)
16	\\(\square\\)	\\(\square^\circ\\)

(simplify your answers. do not include the degree symbol in your answers.)

Explanation:

To solve the problem, we use the formulas for the number of triangles formed in a polygon and the interior angle sum of a polygon.

Formula 1: Number of Triangles Formed in a Polygon

For a polygon with \( n \) sides, the number of triangles formed by drawing diagonals from one vertex is \( n - 2 \).

Formula 2: Interior Angle Sum of a Polygon

The interior angle sum of a polygon with \( n \) sides is \( (n - 2) \times 180^\circ \).

Step 1: Polygon with 12 sides

• Number of Triangles Formed: Given as 10 (which matches \( 12 - 2 = 10 \)).
• Interior Angle Sum: \( (12 - 2) \times 180 = 10 \times 180 = 1800 \).

Step 2: Polygon with 13 sides

• Number of Triangles Formed: \( 13 - 2 = 11 \) (matches the interior angle sum \( 1980^\circ \) since \( 11 \times 180 = 1980 \)).

Step 3: Polygon with 14 sides

• Number of Triangles Formed: Given as 12 (which matches \( 14 - 2 = 12 \)).
• Interior Angle Sum: \( 12 \times 180 = 2160 \).

Step 4: Polygon with 15 sides

• Number of Triangles Formed: \( 15 - 2 = 13 \).
• Interior Angle Sum: \( 13 \times 180 = 2340 \).

Step 5: Polygon with 16 sides

• Number of Triangles Formed: \( 16 - 2 = 14 \).
• Interior Angle Sum: \( 14 \times 180 = 2520 \).

Answer:

Number of Sides of the Polygon	Number of Triangles Formed	Interior Angle Sum
13	11	1980
14	12	2160
15	13	2340
16	14	2520