Answer:
The sum of the interior angles of the polygon is 1260°, and the measure of one interior angle is 140°. The value of x is 20.
Step-by-step explanation:
In a regular polygon, all sides lengths are congruent, and all internal angles are congruent.
The formula for the sum of the interior angles (S) of a regular polygon with n sides is given by:
[tex]S=(n-2)\times 180^{\circ}[/tex]
Since the given regular polygon has 9 sides, the sum of its interior angles is:
[tex]S=(9-2)\times 180^{\circ}[/tex]
[tex]S=7\times 180^{\circ}[/tex]
[tex]S=1260^{\circ}[/tex]
Therefore, the sum of the interior angles of the given polygon is 1260°.
To find the measure of one interior angle (A), divide the sum (S) by 9:
[tex]A=\dfrac{S}{9}[/tex]
[tex]A=\dfrac{1260^{\circ}}{9}[/tex]
[tex]A=140^{\circ}[/tex]
Therefore, the measure of one interior angle of the given polygon is 140°.
Since one internal angle of the given regular polygon is labelled as 7x°, to find the value of x, set 7x° equal to 140° and solve for x:
[tex]7x^{\circ}=140^{\circ}[/tex]
[tex]\dfrac{7x^{\circ}}{7}=\dfrac{140^{\circ}}{7}[/tex]
[tex]x^{\circ}}=20^{\circ}[/tex]
[tex]x = 20[/tex]
Therefore, the value of x is x = 20.
