Question

suppose that the interior angles of a convex pentagon are five numbers, each separated by a value of 1 degree from its neighbors. what is the measure of the third smallest angle?
127.57°
136.97°
131.97°
113.57°
none of these answers are correct.
question 6
1 pts
suppose that the interior angles of a convex pentagon are five numbers, each separated by a value of 1 degree from its neighbors. what is the measure of the largest angle?
______ degrees

Explanation:

Step1: Find the sum of interior angles of a pentagon

The formula for the sum of interior angles of an \(n -\)sided polygon is \((n - 2)\times180^{\circ}\). For \(n = 5\) (a pentagon), the sum \(S=(5 - 2)\times180^{\circ}=540^{\circ}\)

Step2: Let the angles form an arithmetic sequence

Let the smallest angle be \(a_1\), and since the angles form an arithmetic sequence with common difference \(d = 1^{\circ}\), the angles are \(a_1,a_1 + 1,a_1+2,a_1 + 3,a_1+4\)

Step3: Use the sum formula for an arithmetic sequence

The sum of the first \(n\) terms of an arithmetic sequence \(S_n=\frac{n(a_1+a_n)}{2}\), where \(n = 5\) and \(a_n=a_1+(n - 1)d=a_1+4\). So \(S_5=\frac{5(a_1+(a_1 + 4))}{2}=5(a_1 + 2)\)

Step4: Solve for \(a_1\)

Since \(S_5 = 540^{\circ}\), we have \(5(a_1+2)=540\). Divide both sides by \(5\): \(a_1+2 = 108\), then \(a_1=106^{\circ}\)

Step5: Find the largest angle

The largest angle \(a_5=a_1+(5 - 1)d\). Substitute \(a_1 = 106^{\circ}\) and \(d = 1^{\circ}\), we get \(a_5=106+4=110^{\circ}\)

Answer:

\(110\)