Question

the measures of the exterior angles of an octagon are ( x^{circ}, 2x^{circ}, 3x^{circ}, 4x^{circ}, 5x^{circ}, 6x^{circ}, 7x^{circ} ), and ( 8x^{circ} ). solve for ( x ).

Explanation:

Step1: Sum of exterior angles of any polygon

The sum of the exterior angles of any polygon is \(360^{\circ}\).

Step2: Set up the equation

We have the exterior angles \(x^{\circ},2x^{\circ},3x^{\circ},4x^{\circ},5x^{\circ},6x^{\circ},7x^{\circ},8x^{\circ}\).
Sum them up: \(x + 2x+3x + 4x+5x+6x+7x+8x=360\).
Combine like - terms: \((1 + 2+3 + 4+5+6+7+8)x=360\).
Since \(1+2+\cdots+8=\frac{8\times(8 + 1)}{2}=36\) (using the sum formula for an arithmetic series \(S_n=\frac{n(n + 1)}{2}\) where \(n = 8\)), the equation becomes \(36x=360\).

Step3: Solve for \(x\)

Divide both sides of the equation \(36x=360\) by \(36\): \(x=\frac{360}{36}\).

Answer:

\(x = 10\)