Question

8-35. Use your knowledge of polygons to answer the questions below, if possible.

hapter 1

a. How many sides does a polygon have if the sum of the measures of the interior angles is 1980°? 900°?

hapter 2

b. If the exterior angle of a regular polygon is 90°, how many sides does it have? What is another name for this shape?

c. Each interior angle of a regular pentagon has measure 2x + 4°. What is x? Explain how you found your answer.

hapter 3

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d. The measures of four of the exterior angles of a pentagon are 57, 74, 56, and 66. What is the measure of the remaining angle?

e. Find the sum of the interior angles of an 11-gon. Does it matter if it is regular or not?

Answer 1

Answer:

(a)I. 13 Sides II. 7 Sides

(b) 4 Sides, Square

(c)x=52 degrees

(d)x=107 degrees

(e)1620 degrees

Step-by-step explanation:

(a)The Sum of the Interior angle of polygon with n sides is derived using the formula: (n-2)180.

I. If the Interior angle is $1980^0$

Then:

$(n-2)180^0=1980^0\\ Divide both sides by 180^0 to isolate n \\n-2=11\\ Add 2 to both sides of the equation\\n-2+2=11+2\\n=13$

The polygon has 13 sides.

II. If the Interior angle is $900^0$

Then:

$(n-2)180^0=900^0\\ Divide both sides by 180^0 to isolate n \\n-2=5\\ Add 2 to both sides of the equation\\n-2+2=5+2\\n=7$

The polygon has 7 sides.

(b)The sum of the exterior angle of a polygon is 360 degrees,

Each exterior angle of a n-sided regular polygon is: $\frac{360^0}{n}$

If the exterior angle of a regular polygon is 90°

Then:

$90\°=\frac{360^0}{n}\\ 90n=360\\n=4$

The regular polygon has 4 sides and it is called a Square.

(c)The Sum of the Interior angle of polygon with n sides is derived using the formula: (n-2)180.

Each Interior angle of a regular n-sided polygon is:  $\frac{(n-2)180^0}{n}$

For a pentagon, n=5

Then:

$\frac{(5-2)180^0}{5}=2x+4\\108=2x+4\\108-4=2x\\104=2x\\x=52^0$

(d)The sum of the exterior angle of a polygon is 360 degrees.

If four of the exterior angles of a pentagon are 57, 74, 56, and 66.

Let the fifth angle=x

Then:

$57+74+56+66+x=360^0\\253+x=360\\x=107^0$

(e)The Sum of the Interior angle of polygon with n sides is derived using the formula: (n-2)180.

In an 11-gon., n=11

Therefore, the sum of the interior angle=(11-2)180=$1620^0$

The sum of the interior angle does not change either in a regular or irregular polygon.