Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
There are four triangle congruence shortcuts: SSS, SAS, ASA, and AAS. We have triangle similarity if (1) two pairs of angles are congruent (AA) (2) two pairs of sides are proportional and the included angles are congruent (SAS), or (3) if three pairs of sides are proportional (SSS). Notice that AAA, AAS, and ASA are not listed -- to include them would be redundant since they all have two congruent angles.
If you were to prove that two triangles are similar, we're going to draw a comparison with congruence, something that we talked about previously. We said that there were 4 shortcuts for proving two triangles congruent. And those 4 shortcuts were angle side angle, side angle side, side side side and angle angle side.
So if you knew just three things about those two triangles, if it's one of these shortcuts, then yes you could say these triangles must be congruent. There are two that did not work. And those were angle angle angle and side side angle. So these these two shortcuts did not give you enough information to say that these two triangles must be congruent because using only three angles you could construct two triangles that are not the same size.
We're going to draw a comparison with similarity. Let's start off by looking at a case where all we know about two triangles is that 2 angles are congruent. But the triangle angle sum, if these two angles are congruent, then the third angle in each of these triangles must be congruent. And yes this would be a shortcut for saying that these 2 triangles must be similar. Which means that corresponding angles are congruent and corresponding sides are proportional. So under our similarity shortcuts, I'm going to use a different marker here. That one's running out. We're going to say that angle angle is a shortcut.
Now notice that I did not write angle angle angle. The reason being is if all you know are 2 angles, that's enough information because that third angle which I guess I could write in has to be congruent as well. So angle angle is a shortcut.
Let's look at a second case. Let's say that all you knew were a side an included angle and another side. And you also knew that the corresponding sides are proportional. Well this would be enough information to say that these two triangles are similar. So we're going to include side angle side into our list of similarity shortcuts. And last let's say, if all we knew were that 3 sides of 2 different triangles, that correspond, are proportional. So we could write out this proportion that that is constant between corresponding sides. These 2 triangles would also have to be congruent. So we're going to say that side side side is also a shortcut.
Now if I were to compare these two lists, you're going to notice that I omitted angle side angle. And that I omitted angle angle side. The reason being is that if you know that these two angles are congruent, then that is just using the angle angle shortcut. Same thing goes with the angle angle side shortcut. As long as you know that two angles are congruent, that's all I need to know for similarity.
Now the one that you will notice is not on this list is side side angle. So this is kind of like the odd man out for congruence and similarity. This is not enough information to say that 2 triangles must be similar. So you only have three similarity shortcuts that you need to memorize. Angle angle, side, oh no I kind of erased that there. Angle angle, side angle side and side side side.