Question

the sum of the measures of the interior angles of a polygon is given. determine the number of sides for each polygon.
31 (1620^circ)
(180(n - 2) = 1620)
(180n - 360 = 1620)
(180n = 1980)
(n = 11)
11 sides
32 (1800^circ)
33 (540^circ)
34 (1260^circ)
35 (3780^circ)
36 (6840^circ)

Explanation:

Problem 32:

Step1: Start with the formula

The formula for the sum of interior angles of a polygon is \(180(n - 2)\), where \(n\) is the number of sides. We set this equal to \(1800^\circ\):
\(180(n - 2)=1800\)

Step2: Distribute 180

Using the distributive property, we get:
\(180n - 360 = 1800\)

Step3: Solve for \(180n\)

Add 360 to both sides of the equation:
\(180n=1800 + 360\)
\(180n=2160\)

Step4: Solve for \(n\)

Divide both sides by 180:
\(n=\frac{2160}{180}\)
\(n = 12\)

Problem 33:

Step1: Start with the formula

Use the formula \(180(n - 2)\) and set it equal to \(540^\circ\):
\(180(n - 2)=540\)

Step2: Distribute 180

Apply the distributive property:
\(180n - 360 = 540\)

Step3: Solve for \(180n\)

Add 360 to both sides:
\(180n=540 + 360\)
\(180n=900\)

Step4: Solve for \(n\)

Divide by 180:
\(n=\frac{900}{180}\)
\(n = 5\)

Problem 34:

Step1: Start with the formula

Set \(180(n - 2)\) equal to \(1260^\circ\):
\(180(n - 2)=1260\)

Step2: Distribute 180

Expand the left - hand side:
\(180n - 360 = 1260\)

Step3: Solve for \(180n\)

Add 360 to both sides:
\(180n=1260 + 360\)
\(180n=1620\)

Step4: Solve for \(n\)

Divide by 180:
\(n=\frac{1620}{180}\)
\(n = 9\)

Problem 35:

Answer:

s:

• Problem 32: \(12\) sides
• Problem 33: \(5\) sides
• Problem 34: \(9\) sides
• Problem 35: \(23\) sides
• Problem 36: \(40\) sides