Question

if the exterior angles of a convex octagon are ( x + 14 ), ( 3x - 3 ), ( 2x + 15 ), ( 3x - 20 ), ( 4x - 3 ), ( 4x - 10 ), ( 6x ), calculate the largest of the eight angles.

96.62°
90.64°
82.38°
85.96°
none of these answers are correct.

Explanation:

Step1: Use the formula for the sum of exterior angles of a polygon

The sum of exterior angles of any convex polygon is \(360^{\circ}\).
So, \((2x + 15)+(3x-20)+(4x - 3)+(4x-10)+(6x)+(3x - 3)+(3x+10)+(x + 14)=360\)

Step2: Combine like terms

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Step3: Solve for \(x\)

Subtract \(3\) from both sides: \(26x=360 - 3=357\)
Then \(x=\frac{357}{26}\approx13.73\)

Step4: Find each angle

1. \(2x + 15=2\times13.73+15=27.46+15 = 42.46^{\circ}\)
2. \(3x-20=3\times13.73-20=41.19-20 = 21.19^{\circ}\)
3. \(4x - 3=4\times13.73-3=54.92-3 = 51.92^{\circ}\)
4. \(4x-10=4\times13.73-10=54.92-10 = 44.92^{\circ}\)
5. \(6x=6\times13.73 = 82.38^{\circ}\)
6. \(3x - 3=3\times13.73-3=41.19-3 = 38.19^{\circ}\)
7. \(3x+10=3\times13.73+10=41.19+10 = 51.19^{\circ}\)
8. \(x + 14=13.73+14 = 27.73^{\circ}\)

Answer:

\(82.38^{\circ}\)