interior angle of a regular polygon of N sides is equal to the exterior angle of a regular polygon of N sides then number of such pairs n,k is
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Answered by
7
It is 8.
The regular polygon is an octagon.
There is a geometry plus algebra way to the answer, and there is the farmer’s or carpenter’s approach.
The farmer says “to make a stop sign out of a square piece of sheet metal, you draw four equal squares on the corner, and cut off the corners along the diagonals of the square. The 45 degree angles in the half squares cut off are the exterior angles of the octagon. The 45 degree angle left in the square you drew, plus the right angle on the other side form the interior angle, that is 90 degrees larger than the exterior angle.
The geometry plus algebra approach, should not require memorizing formulas, but requires understanding what the terms mean.
The sum of the exterior angles of a convex polygon is 360 degrees is a fact to be recognized. The exterior angle of a regular polygon with n sides measures 360/n degrees, and the adjacent, supplementary interior angle measures (180–360/n) degrees. So, we write the equation
180 – 360/n = 360/n + 90 , and solve for n.
Multiplying both sides of the equal sign times n, we get
180n - 360 = 360 + 90n
180n - 90n = 360 + 360
90n = 720
n = 720/90
n = 8
“The sum of the exterior angles of a polygon is 360 degrees” is not a formula to memorize, but a logical consequence of the definition of exterior angle. The exterior angle is the change of direction as you turn the corner going around the polygon. If you are driving, and decide to go around the block, you would turn a certain angle (probably 90 degrees) at each corner. That is the exterior angle. By the time you complete a whole turn and are back at the starting point, you have turned 360 degrees, which is the sum of the exterior angles.
“Adjacent angle”, “supplementary angles”, “right angle”, and “flat angle” are terms that we need to be able to through around in geometry courses. The farmer and carpenter understand the ideas behind those terms, even if the may use a different jargon.
The regular polygon is an octagon.
There is a geometry plus algebra way to the answer, and there is the farmer’s or carpenter’s approach.
The farmer says “to make a stop sign out of a square piece of sheet metal, you draw four equal squares on the corner, and cut off the corners along the diagonals of the square. The 45 degree angles in the half squares cut off are the exterior angles of the octagon. The 45 degree angle left in the square you drew, plus the right angle on the other side form the interior angle, that is 90 degrees larger than the exterior angle.
The geometry plus algebra approach, should not require memorizing formulas, but requires understanding what the terms mean.
The sum of the exterior angles of a convex polygon is 360 degrees is a fact to be recognized. The exterior angle of a regular polygon with n sides measures 360/n degrees, and the adjacent, supplementary interior angle measures (180–360/n) degrees. So, we write the equation
180 – 360/n = 360/n + 90 , and solve for n.
Multiplying both sides of the equal sign times n, we get
180n - 360 = 360 + 90n
180n - 90n = 360 + 360
90n = 720
n = 720/90
n = 8
“The sum of the exterior angles of a polygon is 360 degrees” is not a formula to memorize, but a logical consequence of the definition of exterior angle. The exterior angle is the change of direction as you turn the corner going around the polygon. If you are driving, and decide to go around the block, you would turn a certain angle (probably 90 degrees) at each corner. That is the exterior angle. By the time you complete a whole turn and are back at the starting point, you have turned 360 degrees, which is the sum of the exterior angles.
“Adjacent angle”, “supplementary angles”, “right angle”, and “flat angle” are terms that we need to be able to through around in geometry courses. The farmer and carpenter understand the ideas behind those terms, even if the may use a different jargon.
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11
Answer:
8 is correct answer
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