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

4. a. 171.818°

b. 72 sides

c. 60°

Explanation:

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

`\bold{Interior Angle =( (n-2) * 180\°)/(n)}`

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

`\bold{Interior Angle =( (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 = (360\° )/(180\° - Interior \:Angle)}`

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

substituting Value,

`\bold{n =( 360\° )/( 180\° - 175\°) = (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{(360\° )/( n)}`

substituting value,

exterior angle=
`\bold{(360\° )/( 6)=60\°}`

`\hrulefill`

Answer 2

d) 171.8°

e) 72

f) 60°

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}&=((44-2) \cdot 180^(\circ))/(44)\\\\&=(42\cdot 180^(\circ))/(44)\\\\&=(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 ((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&=(360^(\circ))/(\sf Number\;of\;sides)\\\\&=(360^(\circ))/(6)\\\\&=60^(\circ)\end{aligned}`

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