Question

section 3.1 homework

1. how many sides does a polygon have if the sum of its angle measures is 2700°?

1. how many sides does an equiangular polygon have if each interior angle measures 156°?

Explanation:

Problem 13

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

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

Step2: Substitute the given sum into the formula and solve for \( n \).

We are given that \( S = 2700^{\circ} \). So we set up the equation:

$$ (n - 2)\times180=2700 $$

First, divide both sides of the equation by \( 180 \):

$$ n - 2=\frac{2700}{180} $$

$$ n - 2 = 15 $$

Then, add \( 2 \) to both sides of the equation:

$$ n=15 + 2 $$

$$ n = 17 $$

Step1: Recall the formula for the measure of each interior angle of an equiangular polygon.

For an equiangular polygon with \( n \) sides, the measure of each interior angle \( I \) is given by \( I=\frac{(n - 2)\times180^{\circ}}{n} \), where \( n \) is the number of sides.

Step2: Substitute the given interior angle measure into the formula and solve for \( n \).

We are given that \( I = 156^{\circ} \). So we set up the equation:

$$ \frac{(n - 2)\times180}{n}=156 $$

Multiply both sides of the equation by \( n \) to get rid of the denominator:

$$ (n - 2)\times180=156n $$

Expand the left - hand side:

$$ 180n-360 = 156n $$

Subtract \( 156n \) from both sides:

$$ 180n-156n-360=0 $$

$$ 24n-360 = 0 $$

Add \( 360 \) to both sides:

$$ 24n=360 $$

Divide both sides by \( 24 \):

$$ n=\frac{360}{24} $$

$$ n = 15 $$

Answer:

The polygon has 17 sides.

Problem 14