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
Four shortcuts allow students to know two triangles must be congruent: SSS, SAS, ASA, and AAS. Knowing only side-side-angle (SSA) does not work because the unknown side could be located in two different places. Knowing only angle-angle-angle (AAA) does not work because it can produce similar but not congruent triangles.
When you're trying to determine if two triangles are congruent, there are 4 shortcuts that will work. Because there are 6 corresponding parts 3 angles and 3 sides, you don't need to know all of them. We said if you know that 3 sides of one triangle are congruent to 3 sides of another triangle, they have to be congruent. The same is true for side angle side, angle side angle and angle angle side.
The reason why these work is because If I give you 3 sides that are congruent there's only one triangle that you can construct, but there's a darker serious side of Geometry that we don't like to talk about and that's the two that don't work. So let's take a look at the first one which is side side angle. Now part of reason why this is the serial type Geometry is because if you switch around the a you get a square root, but I'm not going to give the gratitude of hearing me say that.
If we start off with this angle, and a side so I'm going to say this is a fixed angle and this is a side that's rigid notice that I drew a ray here and I'm saying that we need to make a triangle here and I'm going to say that this point right here is the center of the circle, so its going to be about a radius of my marker and I'm going to draw in, in dotted lines and again I am not an artist so if we have this circle that is centered at that point, notice that using a radius I can construct two different lines that are congruent so I'm not changing that third side but these two triangles are definitely not congruent. To redraw them we have this obtuse triangle here so we have these angles as being congruent we have this side being congruent and we have this third side that I haven't marked so we have 1, 2, 3 so we have side side angle and then this other larger triangle that I was able to draw where we have these two angles being congruent cause I kept that fixed, this side was fixed so these two sides must be congruent and this third side because it's a radius of this circle this side must also be congruent but notice we've created two triangles that are not necessarily congruent which is why side side angle is not a shortcut.
The second shortcut that doesn't work is angle angle angle, couple of different ways to look at this one. One way is to say well if we were to extend that side and if we're to extend this side I can construct a line that is parallel to this side right here and what I've done is I've created corresponding and congruent angles because we have two parallel lines and this is the transversal and this side is also a transversal and this third angle here would have to be congruent to itself, so to redraw this we have two triangles where the 3 angles are corresponding but they're definitely not congruent so we have a little bit of overlap here but the idea is that these two triangles are definitely not congruent but their angles are all corresponding and congruent. The word that we would use for these is similar. But this is not what we're talking about right now because right now we're saying congruence. These two triangles must be exactly identical so the two shortcuts that don't work angle angle angle because we'll create two triangles that'll have different sizes although they're will have same angles and the second one that doesn't work is the side side angle not only because it's a [IB] but also because we create two different triangles.