If Polygon A has fewer than 10 sides and the sum of the interior angles of polygon A is divisible by 16, how many sides does Polygon A have?
Show Hint
- 4
- 5
- 6
- 7
- 8
The Correct Option is C
Solution and Explanation
Step 1: Use the formula for the sum of interior angles.
The sum of the interior angles of a polygon with \( n \) sides is given by the formula: \[ \text{Sum of interior angles} = 180(n - 2) \] We are told that the sum of the interior angles is divisible by 16, so: \[ 180(n - 2) \mod 16 = 0 \] Step 2: Solve for \( n \).
Simplify the equation: \[ 180(n - 2) = 16k \quad \text{(where \( k \) is an integer)} \] Now, check for values of \( n \) under 10 that satisfy this equation. The sum of the angles for each possible polygon with fewer than 10 sides: For \( n = 6 \): \[ 180(6 - 2) = 180 \times 4 = 720 \] Since \( 720 \mod 16 = 0 \), \( n = 6 \) satisfies the condition. Thus, Polygon A has 6 sides.
Step 3: Conclusion.
The correct answer is (C) 6.
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