A space shuttle has a damaged thermoresistant tile, say, of triangular shape. One side (face) of the tile is made of thermoresistant ceramic, the other face of insulating foam. Through poor communications ("...Houston,... can you hear me?.. I can't hear you...") Case 1. One Side and Two Angles: one side of the tile is 20 inches long, one angle is 54 degrees, another angle is 46 degrees. How many different triangular tiles must be made and delivered to space, to make sure that the damage can be fixed? Case 2. One Angle and Two Sides: one angle is 54 degrees, one side of the tile is 20 inches long, another side is 16 inches long. At least, how many different triangular tiles must be made and delivered to space, to make sure that the damage can be fixed? In both cases, assume that the known angles are interior angles of the triangle. Hint: remember, the tile must fit into the damagehole with foam face in.




ACTIVITY: The class can split into two groups  Shuttle and Houston. Shuttle makes hole(s) and orders the tile(s), Houston makes minimal number of possible tiles for each case and delivers to Shuttle. First, they naturally ignore the difference between the two faces of the tile, so only the upper half of the pictures will give the answer. Then, they will take into account two faces being different (ceramic and foam). Tiles can be cut from cardboard with sides of different colors. Houston and Shuttle together check if all the possible cases are covered. Next, they discuss, what additional piece of information they would like to hear which would make the tile uniquely defined, so they will need to make only one tile. This part of discussion is going to be fun and will bring their points of views "right to the point". Also they discuss other Cases: Case 3. Three sides known (two tiles), Case 4. Three angles known (no tile). What kind of transformations does the tile undergo on its way from Houston to Shuttle (groups)? Translation, rotation in space, in plane, flipping... 
Congruency of triangles is basic to highly sophisticated problems of isometrism for more general polygons and polyhedra. It is very important and not an easy topic. The answer to these questions depends on the type of applications. Many times, it may be confusing. Different textbooks formulate them with different underlying specifics. But it is better to practice in one concrete application rather to have general but confusing philosophy. 