Question

suppose that 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.
question 6
1 pts
how many sides would a regular polygon need to have in order for each exterior angle to measure 60°?
sides

Explanation:

Step1: Calculate the sum of interior angles of a heptagon

The formula for 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

Let the seven consecutive whole - number angles be \(x,x + 1,x + 2,x + 3,x + 4,x + 5,x + 6\).

Step3: Set up an equation for the sum of angles

\(x+(x + 1)+(x + 2)+(x + 3)+(x + 4)+(x + 5)+(x + 6)=900\).
Simplify the left - hand side: \(7x+(1 + 2+3 + 4+5 + 6)=900\).
Since \(1+2 + 3+4+5 + 6=\frac{6\times(6 + 1)}{2}=21\), the equation becomes \(7x+21 = 900\).

Step4: Solve for \(x\)

Subtract 21 from both sides: \(7x=900 - 21=879\). Then \(x=\frac{879}{7}\approx125.57\). But \(x\) is a whole number. Let's check the second problem.

Step5: Use the formula for the measure of an exterior angle of a regular polygon

The formula for the measure of an exterior angle of a regular polygon is \(E=\frac{360^{\circ}}{n}\), where \(E\) is the measure of an exterior angle and \(n\) is the number of sides.
Given \(E = 60^{\circ}\), we set up the equation \(\frac{360^{\circ}}{n}=60^{\circ}\).

Step6: Solve for \(n\)

Cross - multiply: \(60n=360\). Then \(n=\frac{360}{60}=6\).

Answer:

For the second problem (since the first set of options has non - whole number values for angles of a polygon with whole - number angle measures as per the problem statement (consecutive whole numbers)): 6