Question
1.) Essential Question How does the number of sides in convex polygons relate to the sums of the measures of
he exterior and interior angles?
he exterior and interior angles?
Ask by Ford Davies. in the United States
Jan 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
In a convex polygon with
sides:
- The sum of the interior angles is
degrees. - The sum of the exterior angles is always
degrees.
Each interior angle plus its corresponding exterior angle equalsdegrees.
Solution
To understand how the number of sides in convex polygons relates to the sums of the measures of the exterior and interior angles, we can break down the problem step by step.
Step 1: Understanding Interior Angles
For a convex polygon with
sides, the sum of the interior angles can be calculated using the formula:
where
is the sum of the interior angles.
Step 2: Understanding Exterior Angles
The sum of the exterior angles of any convex polygon, regardless of the number of sides, is always:
where
is the sum of the exterior angles.
Step 3: Relationship Between Interior and Exterior Angles
For each interior angle
and its corresponding exterior angle
, the relationship can be expressed as:
This means that each pair of interior and exterior angles at a vertex adds up to
degrees.
Step 4: Total Angles in Terms of Sides
If we consider the total number of angles in the polygon:
- There are
interior angles. - There are
exterior angles.
Thus, we can express the total measures of the angles in terms of the number of sides
:
- Total interior angles:
- Total exterior angles:
Conclusion
The relationship can be summarized as follows:
- The sum of the interior angles increases with the number of sides, specifically by
degrees for each additional side. - The sum of the exterior angles remains constant at
degrees, regardless of the number of sides.
In summary, while the sum of the interior angles of a convex polygon increases with the number of sides, the sum of the exterior angles remains constant. This illustrates a fundamental property of polygons and their angles.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Bonus Knowledge
Every convex polygon has a fascinating connection between the number of its sides and the sums of its angles. For any convex polygon with
sides, the sum of its interior angles is given by the formula
. Meanwhile, the sum of the exterior angles of any convex polygon is always a delightful
, regardless of how many sides it has! This neat consistency is one of the many wonders of geometry.
When figuring out the relationship between the number of sides and the angles, it’s crucial to avoid common pitfalls. One key mistake is mixing up the formulas for interior and exterior angles. Also, remember that the number of sides directly influences the total interior angle sum but not the exterior sum, which will always total
. Stay sharp, and you’ll math like a pro!