Project 1: Construction of 5 Parallelograms
Explanations
B: This construction works because alternate interior angles of two parallel lines are congruent, the side BD and AD' are congruent (parallelogram), thus those two triangle with center/top point C' are congruent. This means that the sides of those triangles are correspondingly equal and thus bisected.
C: Triangle EGF and EHF are congruent because all the sides are made from radii of two similar circles (circles create with equal radius). Thus, the parallelogram EGFH has all congruent sides including the opposite sides.
D: Start by creating diagonal line JI and then finding the midpoint L1. Next rotate a copy of one triangle around that midpoint by alpha. We see that the two triangles are congruent because they land directly on top of one another. This means the corresponding angles and sides are congruent as well.
E: First create line I and then label its intersection with the circle point M (creating radii LM). Make a parallel line to line I (line M). Create another line (P) that goes through both the center of the circle L and line M; label this intersection Q. Make a parallel line of P that intersects with point M. We know that the opposite sides LM and QN are congruent and parallel because we created them.
It works because line LM is a radius with given length. Line M was created as a parallel line to LM. The points Q and N (or segment QN) were created to equal the given length of LM. Thus, LM and QN are congruent and parallel.
A: Line RS was drawn parallel to line PQ and line SQ was drawn parallel to line RP. Thus the opposite sides are parallel to one another.