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SSS and SAS - Problem 2

Teacher/Instructor Brian McCall
Brian McCall

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

If there are only two pairs of sides that are known to be congruent, SSS (side side side) can't be used to determine if two triangles are congruent. However, the angles between those two congruent sides are congruent, then by SAS (side angle side), the triangles are congruent.

Recall that congruent sides and angles are illustrated with having the same number of dashes running through them. Additionally, when stating that the triangles are congruent, it is important to write the names of the triangles so that the congruent parts are in the same position in both names. For example, if examining two adjacent triangles sharing a side WY, WY is clearly congruent to itself. Additionally, if segments XY and YZ are congruent, and angles XYW and ZYW are congruent, then the triangles should be named XYW and ZYW.

Our goal with this problem is to determine, do we have two congruent triangles and if so what’s the short-cut? Well the first step here is to redraw this triangle on the bottom. Something I always tell my students is when you have two triangles that are sharing a side, redraw then it will be easier to compare them.

So I’m going to redraw this triangle on the bottom with WY and Z. I can see basically what I’m doing is kind of flipping Z up, so W is going to stay in the same spot, Y is going to stay on the same spot and Z is going to be this vertex up there.

So now let’s carry over our markings. Well we see that ZY has two marks, so I’m going to put those two marks, and an angle ZYW, so we have ZYW has that marking. So now it’s the question, do we have enough information?

Well all in know right now is the side and an angle so we can’t use side-side-side, we might be able to use side-angle-side, SAS, but we’re missing a side. But what you should notice is that in this triangle right here, so I’m going to erase that triangle since I redrew it, we have this side WY and this other triangle right here we have that same side WY, which means that these two must be congruent. It’s the same segment.

So I’m going to say yes, these two triangles are congruent by siding side and now I have to make sure that my vertices correspond. Well what corresponds to Y? In this triangle we have angle Y and in this triangle we have angle Y, so that’s pretty easy, Y has to be our first vertex.

What corresponds to Angle X? Well X corresponds to Z, so we know that Z has to be our next number and our last letter is W. So we have triangle YXW, this triangle, congruent to triangle YZW, and remember this order does matter because it tells you which vertices correspond, and our short-cut is-side angle-side.

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