Two Of The Exterior Angles Of An N-sided Polygon Are 25 And 26, Three Of Its Interior Angles Are 161. This intriguing geometric scenario invites us to explore the properties of polygons, their interior and exterior angles, and how these measurements relate to each other in determining the polygon's characteristics. In this article, we will analyze the given data, derive the number of sides, and understand the relationships that govern polygon angles, providing an in-depth understanding suitable for students, educators, and geometry enthusiasts alike.
Understanding the Basics of Polygon Angles
What Is a Polygon?
A polygon is a two-dimensional closed figure composed of straight line segments called sides. The sides meet at points known as vertices. Polygons are classified based on the number of sides they have, such as triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and so forth.Interior and Exterior Angles
- Interior Angles: These are the angles formed inside the polygon at each vertex.
- Exterior Angles: These are the angles formed outside the polygon when one side is extended beyond a vertex, and they are supplementary to the interior angles at that vertex.
The sum of the exterior angles of any polygon, regardless of the number of sides, is always:
\[ \text{Sum of exterior angles} = 360^\circ \]
Note: Each exterior angle can be calculated as the supplementary angle of the corresponding interior angle, provided the angles are adjacent.
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Analyzing the Given Data
The problem states:
- Two of the exterior angles are 25° and 26°.
- Three of the interior angles are 161° each.
Our goal is to find:
- The total number of sides (n) of the polygon.
- The remaining interior and exterior angles.
- Validate the consistency of the data.
Step 1: Sum of the Known Exterior Angles
The two known exterior angles sum to:
\[ 25^\circ + 26^\circ = 51^\circ \]
Since the sum of all exterior angles is 360°, the sum of the remaining exterior angles is:
\[ 360^\circ - 51^\circ = 309^\circ \]
Note: The remaining \( n - 2 \) exterior angles sum to 309°, since we already know two.
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Step 2: Relationship Between Interior and Exterior Angles
At each vertex: \[ \text{Interior angle} + \text{Exterior angle} = 180^\circ \]Given three interior angles are 161°, their corresponding exterior angles are:
\[ 180^\circ - 161^\circ = 19^\circ \]
So, three exterior angles are 19° each.
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Step 3: Summing Known Exterior Angles
- Known exterior angles:
- Two are 25° and 26°.
- Three are 19° each.
Remaining exterior angles:
\[ 360^\circ - 108^\circ = 252^\circ \]
Number of remaining exterior angles:
\[ n - 5 \]
Sum of remaining exterior angles:
\[ 252^\circ \]
Thus:
\[ \text{Average of remaining exterior angles} = \frac{252^\circ}{n - 5} \]
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Determining the Number of Sides (n)
Since the sum of all exterior angles is 360°, and we've identified:
- 2 exterior angles: 25°, 26°
- 3 exterior angles: 19° each
- Remaining \( n - 5 \) angles: sum to 252°
Total exterior angle sum:
\[ 25 + 26 + 3 \times 19 + \text{sum of remaining} = 360^\circ \]
which checks out.
Now, to find n:
Total interior angles sum:
\[ (n - 2) \times 180^\circ \]
From the interior angles known:
- Three are 161° each.
- The others are unknown, but their corresponding exterior angles are \( 180^\circ - \text{interior angle} \).
The sum of all interior angles:
\[ \text{Sum} = (n - 2) \times 180^\circ \]
Sum of the three known interior angles:
\[ 3 \times 161^\circ = 483^\circ \]
Sum of their corresponding exterior angles:
\[ 3 \times 19^\circ = 57^\circ \]
Remaining interior angles are:
\[ \text{Remaining interior angles sum} = (n - 3) \times \text{average interior angle} \]
But it might be easier to look at the total sum.
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Step 4: Calculating the Total Sum of Interior Angles
Total sum: \[ (n - 2) \times 180^\circ \]Sum of known interior angles:
\[ 483^\circ \]
Remaining interior angles sum:
\[ (n - 3) \times \text{unknown interior angle} \]
However, since interior and exterior angles at each vertex are supplementary:
\[ \text{Interior angle} + \text{Exterior angle} = 180^\circ \]
At the vertices with known interior angles:
- Interior angle: 161°
- Exterior angle: 19°
At vertices with unknown interior angles:
- Exterior angles: known or to be found
- Interior angles: \( 180^\circ - \text{exterior angle} \)
Total sum of interior angles:
\[ (n - 2) \times 180^\circ \]
Total sum of exterior angles:
\[ 360^\circ \]
Sum of exterior angles we already calculated:
\[ 51^\circ + 3 \times 19^\circ + \text{remaining} = 360^\circ \]
From earlier:
\[ 25^\circ + 26^\circ + 3 \times 19^\circ = 108^\circ \]
Remaining exterior angles sum to:
\[ 360^\circ - 108^\circ = 252^\circ \]
Number of remaining exterior angles:
\[ n - 5 \]
Average of remaining exterior angles:
\[ \frac{252^\circ}{n - 5} \]
Corresponding interior angles for these remaining exterior angles:
\[ \text{Interior angle} = 180^\circ - \text{exterior angle} \]
Total sum of interior angles:
\[ (n - 2) \times 180^\circ = \text{Sum of known interior angles} + \text{Sum of unknown interior angles} \]
Known interior angles:
\[ 3 \times 161^\circ = 483^\circ \]
Unknown interior angles:
\[ (n - 5) \times \left( 180^\circ - \frac{252^\circ}{n - 5} \right) \]
Simplifying:
\[ (n - 5) \times 180^\circ - 252^\circ \]
Total interior angles sum:
\[ 483^\circ + (n - 5) \times 180^\circ - 252^\circ = (n - 2) \times 180^\circ \]
Expressed as:
\[ 483^\circ - 252^\circ + (n - 5) \times 180^\circ = (n - 2) \times 180^\circ \]
\[ 231^\circ + (n - 5) \times 180^\circ = (n - 2) \times 180^\circ \]
Subtract \( (n - 5) \times 180^\circ \) from both sides:
\[ 231^\circ = (n - 2) \times 180^\circ - (n - 5) \times 180^\circ \]
Simplify RHS:
\[ 180^\circ \times [(n - 2) - (n - 5)] = 180^\circ \times (3) = 540^\circ \]
Set equal:
\[ 231^\circ = 540^\circ \]
This is a contradiction, indicating an inconsistency in the initial assumptions or calculations.
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Resolving the Inconsistency and Correct Approach
The previous approach reveals that directly equating sums leads to a contradiction, which suggests that the problem requires a different perspective.
Alternative method:
- Recognize that the sum of all exterior angles is 360°, regardless of the number of sides.
- Since two exterior angles are 25° and 26°, and three are 19°, the total of these five is 108°.
Remaining exterior angles sum:
\[ 360^\circ - 108^\circ = 252^\circ \]
Number of remaining exterior angles:
\[ n - 5 \]
Average remaining exterior angle:
\[ \frac{252^\circ}{n - 5} \]
Corresponding interior angles:
\[ 180^\circ - \text{exterior angle} \]
Total interior angles sum:
\[ (n - 2) \times 180^\circ \]
Sum of
