A: This construction works because line AB was drawn parallel to line CD and line DB was drawn parallel to line CA. Thus, opposite sides are parallel to each other.[br][br]B: This construction works because AIA of two parallel lines are congruent and DB is congruent to line AD' (definition of a parallelogram), therefore the two triangles formed with those lines as bases and C' (the center) as the top point are congruent. The corresponding sides/angles of the triangles are congruent and thus bisected.[br][br]C: Triangles ABD and CBD are congruent because each of the sides of both triangles are formed by the length of the circles' radii. This means that parallelogram ABCD has all sides congruent.[br][br]D: First, create a diagonal of the polygon and find that line's midpoint. Next, make a copy of one of the triangles and rotate it using the slider tool, which allows you to see that the two triangles are indeed congruent. This also means that the corresponding sides and angles are congruent.[br][br]E: First, start with a circle that has a center. Then, create a line through the center and label the point of intersection of the line and the circle (point B). Make another line parallel to to line a (line b). Next, create a line going through the point of intersection (line e) and another point on the circle (D), then create a line parallel to this one, but going through the center. We already know that line AB is congruent to line CD (because we made them). [br]This makes sense because AB has a given length (the radius) and line b was created as parallel to line a, which were both cut by other parallel lines. This means that CD is also equal to the length of AB. CD = parallel and congruent to AB.