Parallelogram in Railing

I noticed that the railing next to some stairs formed a parallelogram.

This parallelogram can be proved as a parallelogram since opposite angles are congruent.

If you draw a line from angle A to angle C, you can show theorem 5-11 which states that the segment that joins the midpoints of two sides of a triangle is 1. parallel to the third side and 2. half as long as the third side.

In the diagram this theorem is proved because 1. EF is parallel to BA which is shown by corresponding angles being congruent and 2. EF=15.60cm which is half as long as the 31.20cm length of BA.

Finally,  the area of a parallelogram is bh.  To find the height I drew an altitude from angle C down and used  sin of angle B

(∼NW5)

Congruent Parallelograms?

Although you may not be able to tell by the picture, these blue and yellow parallelograms are the tops of two led containers for my mechanical pencil. I traced them with help from the construct>parallel line, construct>perpendicular line, and construct>midpoint tools in The Geometer's Sketchpad.
I drew two altitudes (k and n) in order to find the height of each parallelogram. Then, I multiplied each height by its corresponding base and, to my surprise, found these parallelograms to not have equal areas, and therefore not be congruent.
The total area of parallelogram ACDF is 17.16 square centimeters.
(~GC2)

Tennis Court

I was playing tennis this weekend and I realized that geometry included both the wall in the background and the net in the image below.

Each individual square is a parallelogram. With drawings on the Geometer's Sketchpad, I was able to prove theorems 5-1 to 5-8.

The measure of angle C is 106.66 degrees, angle D is 73.34 degrees, angle B is 106.66 degrees, and angle A is 73.34 degrees. Same side interior angles measured 180 degrees, and the total angle measure was 360 degrees. (~WM5)