Question

suppose that the interior angles of a convex heptagon are seven numbers, each separated by a value of 1 degree from its neighbors. wh is the measure of the third smallest angle? o 127.57° o 136.07° o 131.07° o 113.57° o none of these answers are correct.

Explanation:

Step1: Calculate the sum of interior angles

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

Step2: Assume the angles

Let the seven interior angles be \(x - 3d\), \(x - 2d\), \(x - d\), \(x\), \(x + d\), \(x + 2d\), \(x+3d\) (an arithmetic sequence). The sum of these angles is \(7x\).
Since \(7x = 900^{\circ}\), then \(x=\frac{900^{\circ}}{7}\approx128.57^{\circ}\).
The third - smallest angle is \(x - d\). If we assume \(d = 5^{\circ}\) (for simplicity of calculation, and also because we are looking for a value close to the options).
The third - smallest angle \(x - d=\frac{900^{\circ}}{7}-5^{\circ}\approx128.57^{\circ}- 5^{\circ}=123.57^{\circ}\) (not matching).
Let's use another approach.
We know that for a convex heptagon, all interior angles are less than \(180^{\circ}\) and greater than \(0^{\circ}\).
If we assume the angles form an arithmetic sequence \(a_1,a_2,\cdots,a_7\) with \(S_n=\sum_{i = 1}^{7}a_i=(n - 2)\times180^{\circ}=900^{\circ}\) (by the sum formula of an arithmetic sequence \(S_n=\frac{n(a_1 + a_7)}{2}\), and \(a_7=a_1+6d\), \(S_7=\frac{7(a_1+a_1 + 6d)}{2}=7(a_1 + 3d)=900^{\circ}\), \(a_1+3d=\frac{900^{\circ}}{7}\approx128.57^{\circ}\))
Let's check the options:

• Option 1: \(127.57^{\circ}\)

Let \(a_1+2d = 127.57^{\circ}\), and \(a_1+3d=\frac{900^{\circ}}{7}\approx128.57^{\circ}\), then \(d = 1^{\circ}\), \(a_1=125.57^{\circ}\), \(a_7=a_1 + 6d=131.57^{\circ}\) (all angles are between \(0^{\circ}\) and \(180^{\circ}\))

Answer:

\(127.57^{\circ}\)