Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Our experts are available to provide in-depth and trustworthy answers to any questions you may have.
Sagot :
To determine the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides, we can use a well-known formula in geometry. Here’s the step-by-step explanation:
1. Understanding the Problem:
- We need to find a formula that gives us the total sum of the interior angles of a polygon based on the number of its sides.
2. Concept of Polygon Interior Angles:
- A polygon with [tex]\( n \)[/tex] sides (an [tex]\( n \)[/tex]-gon) can be divided into [tex]\( n - 2 \)[/tex] triangles.
- This is because any polygon can be triangulated into [tex]\( n - 2 \)[/tex] non-overlapping triangles.
3. Sum of Angles in Triangles:
- We know the sum of the interior angles of a triangle is [tex]\( 180^\circ \)[/tex].
4. Relating Triangles to the Polygon:
- Since the polygon can be divided into [tex]\( n - 2 \)[/tex] triangles, the sum of the interior angles of the polygon is the sum of the interior angles of these [tex]\( n - 2 \)[/tex] triangles.
5. Calculation:
- Each triangle contributes [tex]\( 180^\circ \)[/tex] to the sum.
- Therefore, the sum is [tex]\( (n - 2) \times 180^\circ \)[/tex].
6. Conclusion:
- The sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula [tex]\( (n-2) \times 180^\circ \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{(n-2) 180^\circ} \][/tex]
1. Understanding the Problem:
- We need to find a formula that gives us the total sum of the interior angles of a polygon based on the number of its sides.
2. Concept of Polygon Interior Angles:
- A polygon with [tex]\( n \)[/tex] sides (an [tex]\( n \)[/tex]-gon) can be divided into [tex]\( n - 2 \)[/tex] triangles.
- This is because any polygon can be triangulated into [tex]\( n - 2 \)[/tex] non-overlapping triangles.
3. Sum of Angles in Triangles:
- We know the sum of the interior angles of a triangle is [tex]\( 180^\circ \)[/tex].
4. Relating Triangles to the Polygon:
- Since the polygon can be divided into [tex]\( n - 2 \)[/tex] triangles, the sum of the interior angles of the polygon is the sum of the interior angles of these [tex]\( n - 2 \)[/tex] triangles.
5. Calculation:
- Each triangle contributes [tex]\( 180^\circ \)[/tex] to the sum.
- Therefore, the sum is [tex]\( (n - 2) \times 180^\circ \)[/tex].
6. Conclusion:
- The sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula [tex]\( (n-2) \times 180^\circ \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{(n-2) 180^\circ} \][/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.