- Keith Besaw
Two triangles are congruent if the three corresponding sides are congruent and the three corresponding angles are congruent. In some cases we know that two triangles are congruent by using only a few of these six checks. For example, it may only necessary to check a corresponding pair of sides, then the next corresponding angles (going either clockwise or counterclockwise), and then the next corresponding sides (going in the same direction) to know that two triangles are congruent. This would be known as the SAS postulate (for Side Angle Side) if it was true. Along with SAS are the possibilities SSS (compare three corresponding sides), AAA (compare three corresponding angles), and five others with three letters. This GeoGebra model can be used to check if one of the letter combinations is a postulate. The model has a triangle ABC which you can make into any possible triangle simply by adjusting the vertex B. In addition, there 4 segments and three angles that can be moved around. 3 of the segments always equal the length of a side of triangle ABC. The fourth can be made into any length. The three angles always have the same measure as angles in triangle ABC. Notice that you can also set the length of one leg of each of these angles. Try moving point B around and see how the segments and angles change. The segments and angles can be moved around by right clicking points of the segments or angles or segments of the angles and dragging or rotating them. To test a possible postulate like SSS: 1) Move the segment with the same length as AC down to an open spot next to triangle ABC. Let’s call this segment 1. Notice that segment 1 can’t be rotated to make things easier. 2) Move the segment with the same length as AB down and place it so one end touches an end of segment 1. Call this segment 2. 3) Move the segment with the same length as BC down and place it so one end touches the other end of segment 1. Call this segment 3. 4) Can you rotate the unattached ends of segments 2 and 3 to make a triangle that isn’t identical to triangle ABC? If not, does that mean SSS is a postulate? Use other combinations of angle and segment pieces to check other 3 letter combinations. A couple of notes: you can use the angle congruent to angle A to match both angle A and side AC by making the length of one leg equal to the length of AC. Also, you can use the segment that can be any length to check side lengths or to complete a side when one of the legs of an angle is too short. See YouTube video http://youtu.be/uCmUHxuI7Xs for more information.
1) Write down all 8 possible 3 letter combinations of S and A such as SSS, SAS, and AAS. This are possible triangle congruence postulates like SSS or side-side-side. 2) Which of these really the same as another except that one goes clockwise when the other goes counterclockwise? Does clockwise or counterclockwise make a difference? 3) Which of these can be considered postulates that guarantee two triangles are congruent and which don’t guarantee that two triangles are congruent? 4) Is it possible to check just two things like SS (just check two pairs of corresponding sides)? 5) For those 3 letter cases which aren’t postulates, can we make them into 4 letter postulates by adding an S or A either before or after the 3 letters? Do these 4 letter postulates already contain one of the 3 letter postulates as a subset? 6) For those 3 letter cases which aren’t postulates, can we add special conditions that turn them into postulates like requiring the triangles to be scalene, obtuse, or a right or requiring the measures of sides or angles to be ordered in some way?