Question

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

a. The sum of the interior of a 73-gon.

b. The number of sides of a regular polygon if the sum of the interior angles is 2700°

c. The number of sides of a reguluar polygon if the exterior angle is 7.2°

Answer 1

Answer:

see explanation

Step-by-step explanation:

a

the sum of the interior angles of a polygon is calculated as

sum = 180° (n - 2) ← n is the number of sides

here n = 73 , then

sum = 180° × (73 - 2) = 180° × 71 = 12,780°

b

here sum of interior angles = 2700° , then

180° (n - 2) = 2700° ( divide both sides by 180° )

n - 2 = 15 ( add 2 to both sides )

n = 17

number of sides is 17

c

the sum of the exterior angles of a polygon = 360°

given the polygon is regular then each exterior angle is congruent , so

number of sides = 360° ÷ 7.2° = 50

 

Answer 2

Answer:

a)  12870°

b)  17

c)  50

Step-by-step explanation:

Part a

The Polygon Interior Angle-Sum Theorem states that the sum of the measures of the interior angles of a polygon with n sides is (n - 2) · 180°.

The number of sides of a 73-gon is n = 73. Therefore, the sum of its interior angles is:

$\begin{aligned}\textsf{Sum of the interior angles of a 73-agon}&=(73-2) \cdot 180^{\circ}\\&=71 \cdot 180^{\circ}\\&=12870^{\circ}\end{aligned}$

Therefore, the sum of the interior angles of a 73-gon is 12870°.

$\hrulefill$

Part b

The Polygon Interior Angle-Sum Theorem states that the sum of the measures of the interior angles of a polygon with n sides is (n - 2) · 180°.

Given the sum of the interior angles of a regular polygon is 2700°, then:

$\begin{aligned} \textsf{Sum of the interior angles}&=2700^{\circ}\\\\\implies (n-2) \cdot 180^{\circ}&=2700^{\circ}\\n-2&=15\\n&=17\end{aligned}$

Therefore, the number of sides of the regular polygon is 17.

$\hrulefill$

Part c

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

Therefore, to find the number of sides of a regular polygon given its exterior angle is 7.2°, divide 360° by the exterior angle.

$\begin{aligned}\textsf{Number of sides}&=\dfrac{360^{\circ}}{\sf Exterior\;angle}\\\\&=\dfrac{360^{\circ}}{7.2^{\circ}}\\\\&=50\end{aligned}$

Therefore, the number of sides of the regular polygon is 50.