Question

question 1-13
tammy was asked to determine if a regular 24 - gon would tessellate the plane. tammy used the formula to determine the interior angle of a regular polygon
tammy determined that because

a factor of 360° a regular 24 - gon

tessellate the plane.

Explanation:

Step 1: Recall the formula for the interior angle of a regular polygon

The formula for the measure of an interior angle of a regular \( n \)-gon is \( I=\frac{(n - 2)\times180^{\circ}}{n} \). For \( n = 24 \), we substitute \( n \) into the formula: \( I=\frac{(24-2)\times180^{\circ}}{24}=\frac{22\times180^{\circ}}{24}=\frac{3960^{\circ}}{24} = 165^{\circ} \).

Step 2: Check if the interior angle is a factor of \( 360^{\circ} \)

We need to see if \( 360^{\circ}\div165^{\circ} \) gives an integer. \( 360\div165=\frac{24}{11}\approx2.18 \), which is not an integer. So the interior angle of a regular 24 - gon is not a factor of \( 360^{\circ} \).

Step 3: Determine if the polygon tessellates

A regular polygon tessellates the plane if and only if the measure of its interior angle is a factor of \( 360^{\circ} \). Since \( 165^{\circ} \) is not a factor of \( 360^{\circ} \), a regular 24 - gon does not tessellate the plane.

So the first blank should be filled with the measure of the interior angle of the regular 24 - gon (which is \( 165^{\circ} \)), the second blank (the relationship) is "is not", and the third blank is "does not".

Answer:

First blank: \( 165^{\circ} \); Second blank: is not; Third blank: does not