Question

suppose the measures of the interior angles of a convex heptagon are seven consecutive whole numbers. find the measure of the fifth angle. 131.57° 126.26° 123.4° 129.57° none of these answers are correct.

Explanation:

Step1: Recall the formula for the sum of interior angles of a polygon

The sum of interior angles of an \(n -\)sided polygon is \((n - 2)\times180^{\circ}\). For a heptagon (\(n=7\)), the sum \(S=(7 - 2)\times180^{\circ}=900^{\circ}\).

Step2: Let the seven consecutive whole - number angles be \(x - 3,x - 2,x - 1,x,x + 1,x + 2,x + 3\)

The sum of these seven angles is \((x-3)+(x - 2)+(x - 1)+x+(x + 1)+(x + 2)+(x + 3)=7x\).

Step3: Solve for \(x\)

Since \(7x = 900\), then \(x=\frac{900}{7}\approx128.57\). But if we assume the angles are \(a,a + 1,a + 2,a+3,a + 4,a + 5,a + 6\)
The sum \(S=\sum_{i = 0}^{6}(a + i)=7a+\sum_{i = 0}^{6}i=7a+\frac{6\times(6 + 1)}{2}=7a+21\)
We know \(7a+21=900\), so \(7a=879\), \(a=\frac{879}{7}\approx125.57\).
Another way: Let the middle - valued angle (the fourth angle) be \(x\). The sum of seven consecutive whole - number angles is \(7x\) (because \((x-3)+(x - 2)+(x - 1)+x+(x + 1)+(x + 2)+(x + 3)=7x\)). Since \(7x=900\) (sum of interior angles of heptagon), \(x=\frac{900}{7}\approx128.57\). If we assume the angles are \(x-2,x - 1,x,x + 1,x + 2,x+3,x + 4\), sum is \(7x+7\). If \(7x + 7=900\), \(x=\frac{893}{7}\approx127.57\). If the angles are \(x-1,x,x + 1,x + 2,x + 3,x+4,x + 5\), sum is \(7x+14\). If \(7x+14 = 900\), \(x=\frac{886}{7}\approx126.57\).

Answer:

None of these answers are correct.