Uncover valuable information and solutions with IDNLearn.com's extensive Q&A platform. Get step-by-step guidance for all your technical questions from our dedicated community members.
Sagot :
To determine the sum of the measures of the interior angles of a regular polygon where each exterior angle measures [tex]\( 120^\circ \)[/tex], follow these steps:
1. Understanding Exterior Angles of a Polygon:
The exterior angles of any polygon always add up to [tex]\( 360^\circ \)[/tex].
2. Find the Number of Sides:
The measure of each exterior angle is given as [tex]\( 120^\circ \)[/tex]. The number of sides of the polygon can be found by dividing [tex]\( 360^\circ \)[/tex] by the measure of each exterior angle:
[tex]\[ \text{Number of sides} = \frac{360^\circ}{120^\circ} = 3 \][/tex]
So, the polygon has 3 sides and is a triangle.
3. Calculate the Sum of the Interior Angles:
The formula to find the sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is:
[tex]\[ \text{Sum of the interior angles} = (n - 2) \times 180^\circ \][/tex]
For a triangle ([tex]\( n = 3 \)[/tex]):
[tex]\[ \text{Sum of the interior angles} = (3 - 2) \times 180^\circ = 1 \times 180^\circ = 180^\circ \][/tex]
Therefore, the sum of the measures of the interior angles of this polygon is:
[tex]\[ 180^\circ \][/tex]
So, the correct answer is [tex]\( \boxed{180^\circ} \)[/tex].
1. Understanding Exterior Angles of a Polygon:
The exterior angles of any polygon always add up to [tex]\( 360^\circ \)[/tex].
2. Find the Number of Sides:
The measure of each exterior angle is given as [tex]\( 120^\circ \)[/tex]. The number of sides of the polygon can be found by dividing [tex]\( 360^\circ \)[/tex] by the measure of each exterior angle:
[tex]\[ \text{Number of sides} = \frac{360^\circ}{120^\circ} = 3 \][/tex]
So, the polygon has 3 sides and is a triangle.
3. Calculate the Sum of the Interior Angles:
The formula to find the sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is:
[tex]\[ \text{Sum of the interior angles} = (n - 2) \times 180^\circ \][/tex]
For a triangle ([tex]\( n = 3 \)[/tex]):
[tex]\[ \text{Sum of the interior angles} = (3 - 2) \times 180^\circ = 1 \times 180^\circ = 180^\circ \][/tex]
Therefore, the sum of the measures of the interior angles of this polygon is:
[tex]\[ 180^\circ \][/tex]
So, the correct answer is [tex]\( \boxed{180^\circ} \)[/tex].
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.