Question

find the sum of the interior angle measures of the polygon. (see example 1.)
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1. smp3 you be the teacher your friend finds the sum of the interior angle measures of a 13 - gon.

is your friend correct? explain your reasoning.

find the value of x. (see example 2.)
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1. smp3 you be the teacher your friend finds the measure of each interior angle of a regular 20 - gon. is your friend

correct? explain your reasoning.

1. connecting to real life the border

of a susan b. anthony dollar is a regular
polygon. what is the measure of each
interior angle of the border? round your
answer to the nearest degree. (see example 3.)

Explanation:

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

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

Step2: Analyze friend's work for the 13 - gon

Your friend used \(S=n\times180^{\circ}\). For a polygon, the correct formula is \(S=(n - 2)\times180^{\circ}\). When \(n = 13\), the correct sum is \(S=(13- 2)\times180^{\circ}=11\times180^{\circ}=1980^{\circ}\), not \(13\times180^{\circ}=2340^{\circ}\).

Step3: Analyze friend's work for the 20 - gon

Your friend correctly found the sum of interior angles \(S=(n - 2)\times180^{\circ}\) for \(n = 20\), \(S=(20 - 2)\times180^{\circ}=3240^{\circ}\). But for a regular 20 - gon, the measure of each interior angle is \(\frac{(n - 2)\times180^{\circ}}{n}\), not \(\frac{S}{n-2}\). The correct measure is \(\frac{(20 - 2)\times180^{\circ}}{20}=\frac{3240^{\circ}}{20}=162^{\circ}\), not \(180^{\circ}\).

Answer:

1. Your friend is incorrect. The formula for the sum of interior angles of a polygon is \(S=(n - 2)\times180^{\circ}\), not \(S=n\times180^{\circ}\). For \(n = 13\), \(S=(13 - 2)\times180^{\circ}=1980^{\circ}\).
2. Your friend is incorrect. For a regular \(n\) - gon, the measure of each interior angle is \(\frac{(n - 2)\times180^{\circ}}{n}\). For \(n = 20\), \(\frac{(20 - 2)\times180^{\circ}}{20}=162^{\circ}\), not \(180^{\circ}\).