Question

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4. Use the theorems for interior and exterior angles of a polygon to find:

d. The interior angle of a regular 44-gon.

e. The number of sides in a regular polygon with an interior angle is 175°.

f. The exterior angle in a regular hexagon.

Answer 1

Answer:

d)  171.8°

e)  72

f)  60°

Step-by-step explanation:

Part d

The Polygon Interior Angle Theorem states that measure of the interior angle of a regular polygon with n sides is [(n - 2) · 180°] / 2.

The number of sides of a 44-gon is n = 44. Therefore, the measure of its interior angle is:

$\begin{aligned}\textsf{Interior angles of a 44-agon}&=\dfrac{(44-2) \cdot 180^{\circ}}{44}\\\\&=\dfrac{42\cdot 180^{\circ}}{44}\\\\&=\dfrac{7560^{\circ}}{44}\\\\&=171.8^{\circ}\;\sf(nearest\;tenth)\end{aligned}$

Therefore, the interior angle of a 44-gon is 171.8°.

$\hrulefill$

Part e

The Polygon Interior Angle Theorem states that measure of the interior angle of a regular polygon with n sides is [(n - 2) · 180°] / 2.

Given the interior angle of a regular polygon is 175°, then:

$\begin{aligned} \textsf{Interior angle}&=175^{\circ}\\\\\implies \dfrac{(n-2) \cdot 180^{\circ}}{n}&=175^{\circ}\\\\(n-2)\cdot 180^{\circ}&=175^{\circ}n\\\\180^{\circ}n-360^{\circ}&=175^{\circ}n\\\\5^{\circ}n&=360^{\circ}\\\\n&=72\end{aligned}$

Therefore, the number of sides of the regular polygon is 72.

$\hrulefill$

Part f

According the the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles of a polygon is 360°.

Therefore, to find exterior angle of a regular hexagon, divide 360° by the number of sides:

$\begin{aligned}\sf Exterior\;angle&=\dfrac{360^{\circ}}{\sf Number\;of\;sides}\\\\&=\dfrac{360^{\circ}}{6}\\\\&=60^{\circ}\end{aligned}$

Therefore, the exterior angle of a regular hexagon is 60°.

Answer 2

Answer:

4. a. 171.818°

b. 72 sides

c. 60°

Step-by-step explanation:

d. The interior angle of a regular polygon with n sides can be calculated using the formula:

$\bold{Interior Angle =\frac{ (n-2) * 180\°}{n}}$

For a regular 44-gon, the interior angle would be:

$\bold{Interior Angle =\frac{ (44-2) * 180\°}{44}=171.818^o}$

$\hrulefill$

e. The number of sides in a regular polygon with an interior angle of 175° can be found using the formula:

$\bold{n = \frac{360\° }{180\° - Interior \:Angle}}$

For an interior angle of 175°, the number of sides would be:

substituting Value,

$\bold{n =\frac{ 360\° }{ 180\° - 175\°} = \frac{360\° }{ 5\° }= 72 \:sides}$

$\hrulefill$

f.

The sum of the exterior angles of any polygon is always 360°.

Since a regular hexagon has six sides,
n=6

exterior angle would be $\bold{\frac{360\° }{ n}}$

substituting value,

exterior angle=$\bold{\frac{360\° }{ 6}=60\°}$

$\hrulefill$