How many sides has a polygon if the sum of its interior angles equals the sum of its exterior angles?
The correct answer and explanation is :
To find how many sides a polygon has if the sum of its interior angles equals the sum of its exterior angles, we need to understand the relationships between these angles in a polygon.
Step 1: Understand the sum of interior angles
For any polygon with (n) sides, the sum of the interior angles is given by the formula:
[
\text{Sum of interior angles} = 180(n – 2)
]
where (n) is the number of sides of the polygon.
Step 2: Understand the sum of exterior angles
The sum of the exterior angles of any polygon is always 360 degrees, regardless of the number of sides. This holds true because each exterior angle is formed by extending one side of the polygon and measuring the angle between the extended side and the adjacent side. The exterior angles always sum to 360°.
Step 3: Set up the equation
We are told that the sum of the interior angles equals the sum of the exterior angles. From the previous steps:
[
180(n – 2) = 360
]
Step 4: Solve the equation
Now, solve for (n):
[
180(n – 2) = 360
]
Divide both sides by 180:
[
n – 2 = 2
]
Now, solve for (n):
[
n = 4
]
Step 5: Interpret the result
The polygon must have 4 sides. A polygon with 4 sides is a quadrilateral.
Step 6: Verify the solution
For a quadrilateral ((n = 4)):
- The sum of the interior angles is (180(4 – 2) = 360) degrees.
- The sum of the exterior angles is always 360 degrees.
Since both sums are equal, the solution is correct.
Conclusion:
The polygon must be a quadrilateral with 4 sides.