Question

pentagon 5
hexagon 6
heptagon 7
octagon 8
nonagon 9
decagon 10
n - gon n n - 2 (n - 2)180°

Explanation:

Step1: Recall polygon - triangle relationship

The number of non - overlapping triangles formed by diagonals from one vertex of an $n$-sided polygon is $n - 2$.

Step2: Recall sum of interior angles formula

The sum of the interior angles of an $n$-sided polygon is $(n - 2)\times180^{\circ}$ because each triangle has an interior - angle sum of $180^{\circ}$.
For a pentagon ($n = 5$):

• The number of non - overlapping triangles formed by diagonals from one vertex is $5-2=3$.
• The sum of the interior angles is $(5 - 2)\times180^{\circ}=540^{\circ}$.

For a hexagon ($n = 6$):

• The number of non - overlapping triangles formed by diagonals from one vertex is $6 - 2=4$.
• The sum of the interior angles is $(6 - 2)\times180^{\circ}=720^{\circ}$.

For a heptagon ($n = 7$):

• The number of non - overlapping triangles formed by diagonals from one vertex is $7 - 2=5$.
• The sum of the interior angles is $(7 - 2)\times180^{\circ}=900^{\circ}$.

For an octagon ($n = 8$):

• The number of non - overlapping triangles formed by diagonals from one vertex is $8 - 2=6$.
• The sum of the interior angles is $(8 - 2)\times180^{\circ}=1080^{\circ}$.

For a nonagon ($n = 9$):

• The number of non - overlapping triangles formed by diagonals from one vertex is $9 - 2=7$.
• The sum of the interior angles is $(9 - 2)\times180^{\circ}=1260^{\circ}$.

For a decagon ($n = 10$):

• The number of non - overlapping triangles formed by diagonals from one vertex is $10 - 2=8$.
• The sum of the interior angles is $(10 - 2)\times180^{\circ}=1440^{\circ}$.

The completed table:

Polygon	Number of Sides ($n$)	Number of Non - overlapping Triangles from One Vertex ($n - 2$)	Sum of Interior Angles ($(n - 2)180^{\circ}$)
hexagon	6	4	$720^{\circ}$
heptagon	7	5	$900^{\circ}$
octagon	8	6	$1080^{\circ}$
nonagon	9	7	$1260^{\circ}$
decagon	10	8	$1440^{\circ}$

Answer:

Polygon	Number of Sides ($n$)	Number of Non - overlapping Triangles from One Vertex ($n - 2$)	Sum of Interior Angles ($(n - 2)180^{\circ}$)
hexagon	6	4	$720^{\circ}$
heptagon	7	5	$900^{\circ}$
octagon	8	6	$1080^{\circ}$
nonagon	9	7	$1260^{\circ}$
decagon	10	8	$1440^{\circ}$