High School: Geometry
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow the definition of congruence in terms of rigid motions.
Basically, this standard requires students to fall in love with triangles. They should be so head-over-heels that triangles become tri-angels. (That's an angel that's three times as heavenly.)
Of course, falling in love with triangles isn't easy. If your students who think geometry is a four-letter word, they may not want to enter in any spelling bees in the near future. But more than that, they can use three-letter words to save themselves from drowning in the oddly symmetrical sea of geometric congruence.
Specifically, we're talking about the three-letter words SSS, SAS, and ASA. They're actually acronyms for "Side Side Side," "Side Angle Side," and "Angle Side Angle." But we don't judge.
These 3 acronyms represent different combinations of sides and angles that must correspond in order for two triangles to be congruent: all corresponding sides, two corresponding sides and their included angles, or two corresponding angles and any corresponding side. If two triangles meet just one of these three criteria, they're congruent.
Students should know not only what these three rules are and how they work, but also when to use which one based on the situation. Sometimes more than one is applicable!
We also recommend showing why these rules work. A good way to do this is to find all angles and side lengths of two triangles while only given enough to satisfy the SSS, SAS, or ASA rules. Students should understand that when even one of these rules applies, the triangles are congruent not because the rule "says so," but because it's a quick way of realizing that all side lengths and angles are congruent.
Basically, they should understand that the concept of congruence hasn't changed. You've just given them shortcuts. All the more reason to fall in love with triangles, we think. In fact, don't be surprised if your students have already picked a date for the wedding.
Need a recap? Here is a video to help you out on ASA, SAS, SAS....