Question 524132: There is a famous theorem in Euclidean geometry that states that the sum of the interior angles of a triangle is 180 degrees.
a)Use the theorem about triangles to determine the sum of the angles of a convex quadrilateral. Hint: Draw a convex quadrilateral and draw a diagonal.
b) Use the result in Part(1) to determine the sum of the angles of a convex pentagon.
c) Use the result in Part(2) to determine the sum of the angles of a convex hexagon.
d)Let 'n' be a natural number with 'n'> or = to 3. Make a conjecture about the sum of the angles of a convex polygon with 'n' sides and use mathematical induction to prove your conjecture.
** I've figured out a, b, and c. I've also created a conjecture for d. Here's what I have: "Let 'n' be a natural number with 'n'> or = to 3. For any convex polygon with 'n' sides, the sum of the angle of the polygon is 180(n-2)". I have my basis class which is when n=3 then the sum = 180 degrees, but I don't know how to prove my induction step. I know (k+1) must be substitued in for 'n' at some point but not sure when and what to do. Please help.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Make a conjecture about the sum of the angles of a convex polygon with 'n' sides and use mathematical induction to prove your conjecture.
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Prove that an n sided quadrilateral
has sum of interior angles = (n-2)180 degrees
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Step #1:
For n = 3, sum of angles = (3-2)180 = 180::::that is true
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Step # 2:
Assume true for n = k sides, i.e,
sum of interio angles = (k-2)180
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Step # 3:
Prove true for n = k+1 sides
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sum for k+1 = sum for k sides + 180
sum for k+1 = (k-2)180 + 180
Factor out the 180 to get:
sum for k+1 = (k-2+1)(180)
Therefore, sum for k+1 sides = [(k+1)-2)]180
Q.E.D.
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