We have shown in class that the sum of the interior angles of a triangle is 180 degrees in Euclidean geometry. Use this fact to answer each of the questions below about convex polygons. For each part of the problem, answer the questions, draw a picture and write 1-3 sentences justification for your solution.
(a) Do all quadrilaterals have the same sum of interior angles? What is the sum of the interior angles of a quadrilateral?
(b) Do all pentagons have the same sum of interior angles? What is the sum of the interior angles of a pentagon?
(c) Do all hexagons have the same sum of interior angles? What is the sum of interior angles of a hexagon?
(d) Using your answers to the previous part of the problem, predict whether the sum of interior angles in an octagon is always the same, and if so, what that sum should be. Then, test your prediction.
(e) Generalize: Write a conjecture for the sum of interior angles of a convex $n$-gon in terms of $n$.
(f) Prove your conjecture.