We need a way of describing when two shapes are exactly the same. The word we use in geometry is “congruent.” Two shapes are congruent if they are exact copies of one another, that is, if they can be placed one over they other so that they coincide. You may remember studying a related idea, called similarity, in the first weeks of our class. You should be careful to understand the difference between these two words: congruence means same size and shape, while similarity means same shape but not necessarily same size.
We will be working with relatively simple shapes, triangles, and discussing various methods for showing that two triangles are congruent. Congruence says that all the corresponding parts must be equal. When dealing with triangles, there are six parts to consider: three angles, and three sides. If we want to show that two triangles are congruent, one way to do it is to consider each of these six parts one at a time, showing that all corresponding sides and corresponding angles are equal. However, in practice, we don’t have to consider all six of these parts — usually we can get away with only a few of them. Which ones? We will learn a number of facts (or theorems) that tell us which combinations of sides and angles suffices to show that two triangles are equal — one example is “side angle side (SAS).”
The following video gives a detailed introduction to congruence of triangles. It concludes with another theorem (which we will talk more about next time) called SSS or “side side side”.
The theorem SAS, or “side angle side”, says that if two sides and their included angle in a triangle (the included angle is the angle that lies between the two sides, not one of the other angles in the triangle) are respectively equal to two sides and their included angle in another triangle, then the two triangles are congruent. This is useful, because it says we only have to check three out of the six parts – two sides and the angle between them – in order to conclude that the triangles are congruent.
The next video gives an explanation of the SAS theorem. It then provides a detailed example of how we use the theorem to solve a problem (and show two triangles are congruent). You will often have to pull in some extra information — in this video, they use facts about parallel lines that we covered at the beginning of the semester (the theorem they call AIA, or Alternate Interior Angles, we referred to as the “Z Theorem”). NOTE: the example is cut off at the end, and continues in the second video below.
The next video is the continuation of the previous one. After completing the example they go on to other topics which you can ignore (you only need to watch the first 2 minutes or so).