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Answer: 10 sidesInterior angle = (180 * (number_of_sides - 2)) / number_of_sides

Interior angle = 180 * (10-2) / 10 = 180 * 8 / 10 = 144

Exterior angle = 360 / number_of_sides = 360 / 10 = 36 degrees

Ratio (144:36 = 4:1)

Definitions:The exterior angle (e) is the angle created from the side and the imaginary line that would be if the adjacent side were to be extended beyond the vertices. The interior angle (i) plus the exterior angle (e) add up to 180 degrees, so i + e = 180.In a normal polygon, the sum of the exterior angle of all vertices adds up to 360 degrees; the sides do not overlap or cross. In a regular polygon each external angle is the same, and will divide evenly into 360 degrees to give the number of sides (n): 360 / e = n.

Proof:Given that the ratio of the two angles is 4 to 1: i = 4e.Now substitute: i + e = 180; 4e + e = 5e = 180.

e = 180 / 5 = 36 degrees.

i = 180 - e = 144 degrees.

360/e = 360/36 = 10 sides.

The interior angle of a regular normal polygon of n sides is: i = (n-2)*180/n.

For n=10: i = (10-2)*180/10 = 8*18 = 144 degrees, so it checks.

Other i to e ratios:1 to 2: 3 sides: triangle (trigon)2 to 2: 4 sides: tetrahedron (tetragon, square)

3 to 2: 5 sides: pentahedron (pentagon)

4 to 2: 6 sides: hexahedron (hexagon)

5 to 2: 7 sides: heptahedron

6 to 2: 8 sides: octahedron (octagon)

7 to 2: 9 sides: enneahedron (nonogon)

8 to 2: 10 sides: decahedron (decagon)

Q: The ratios of the measures of an interior angle to an exterior angle of a regular polygon is 4 to 1 How many sides does the polygon have?

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1440 degrees

1440 degrees

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If exterior angles are 120 then interior angles must be 60 so you have an equilateral triangle

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Hint: in a regular polygon interior plus exterior = 180o, always.

1260

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1440 degrees

1440 degrees

1800 degrees

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If exterior angles are 120 then interior angles must be 60 so you have an equilateral triangle

The interior angle of any regular polygon can be calculated using the formula 180 * (n - 2) / n, where n is the number of sides. In this case, since each exterior angle measures 72 degrees, the interior angle would be 180 - 72 = 108 degrees. So the measures of the interior angles in this regular polygon would be 108 degrees.

1440 degrees

540 degrees

Interior angle = 120 degrees Exterior angle = 60 degrees