Question

learning target: i can apply the properties of triangles to solve algebraic equations.
determine the value of x in each triangle.
a)
b)
find the indicated angle measures.
solve for x.
learning target: i can apply the properties of triangles to solve angles in polygons.
chelsea drew a 18 - sided polygon.
a) calculate the sum of the interior angles of the figure.
b) suppose the figure is a regular polygon. use the formula to calculate each interior angle measure.
c) what is the sum of the exterior angles of the figure?
d) suppose the figure is a regular polygon. what is the measure of each exterior angle?

Explanation:

Step1: Sum of interior angles formula

The formula for the sum of interior angles of an \(n -\)sided polygon is \(S=(n - 2)\times180^{\circ}\). For \(n = 18\), we substitute \(n\) into the formula.

$$S=(18- 2)\times180^{\circ}$$

Step2: Calculate the sum

First, calculate \(18 - 2=16\). Then, \(16\times180^{\circ}=2880^{\circ}\)

Step3: Measure of each interior angle (regular polygon)

For a regular \(n -\)sided polygon, the measure of each interior angle \(I=\frac{(n - 2)\times180^{\circ}}{n}\). Substituting \(n = 18\), we have \(I=\frac{2880^{\circ}}{18}=160^{\circ}\)

Step4: Sum of exterior angles

The sum of exterior angles of any polygon is \(360^{\circ}\)

Step5: Measure of each exterior angle (regular polygon)

For a regular \(n -\)sided polygon, the measure of each exterior angle \(E=\frac{360^{\circ}}{n}\). Substituting \(n = 18\), we get \(E=\frac{360^{\circ}}{18}=20^{\circ}\)

Answer:

a) The sum of the interior angles is \(2880^{\circ}\)
b) Each interior angle measure is \(160^{\circ}\)
c) The sum of the exterior angles is \(360^{\circ}\)
d) Each exterior angle measure is \(20^{\circ}\)