Similar Triangles with Parallel Sides

 

While listening to music, I noticed that the logo on my headphones is an example of two similar triangles. To prove this, I traced both triangles, and then found the midpoints of lines AC and BC. These two points happened to land precisely where expected, and so I connected them to form a white overlay of the symbol.

I used the measuring tool in GSP to find all five angles, which are listed on the left. By using the AA postulate, we can see that triangle ABC and triangle EDC are indeed similar.

Although it's not included in the picture above, I also measured angle AED. The sum of angles AED and BAC was 180 degrees, which proves that side AB is parallel to side ED.

(~GC2)

Congruent Angles in a Stool

I noticed that a stool in my house had rungs that were parallel to each other. I figured that if the rungs were parallel then the corresponding angles would be congruent. After taking the picture and putting it into GSP, I found that angle BAF was indeed congruent to angle DCF. (~DM2)

Parallel Lines

This shows that angle ABC is congruent to angle ADE, and therefore segment BC is parallel to DE by postulate 11 -- it states that if corresponding angles are congruent when a transversal crosses two lines then those lines are parallel. (~JZ5)