Polygon angle ratio

In a certain polygon, the ratio of the sum of the sizes of its internal angles and the sum of the sizes of the complementary angles is 2:5. How many vertices does this polygon have?

Final Answer:

n =  8

Step-by-step explanation:

s1​:s2​ = 2:5  s1​ = (n−2)⋅ 180  ∘ s1​ = α1​+α2​+α3​+…αn s2​ = (90−α1​)+(90−α2​)+(90−α3​)+…(90−αn) = 90⋅ n − s1​  s1​:s2​ = 2:5 5s1​ = 2 s2​  5⋅ s1​ = 2⋅ (90⋅ n − s1​) 7⋅ s1​ = 2⋅ 90⋅ n  7⋅ (n−2)⋅ 180=2⋅ 90⋅ n  1080n=2520  n=10802520​=2.33333333=8  n=37​≐2.333333  n=8   Verifying Solution:   s1​=(n−2)⋅ 180=(8−2)⋅ 180=1080 ∘ s2​=n⋅ 180−s1​=8⋅ 180−1080=360 ∘ r=s1​:s2​=1080:360=3=3:1

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algebraplanimetrybasic operations and conceptsnumbersUnits of physical quantitiesGrade of the word problem

 
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