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
In order to determine how two triangles are congruent, one must first determine if there is enough information to say whether or not they are. To see if there are congruent angles, use other concepts from geometry. Recall that vertical angles, alternate interior angles, alternate interior angles, and alternate exterior angles are congruent.
By ASA, if two angles and the side between them are congruent two corresponding angles the side between them, then the two triangles are congruent. By AAS, if two angles and a side of a triangle are congruent to two angles and a side of another triangle, the two triangles are congruent. The position of the angles and side is important--the side must either be between the two angles in both triangles, or beside one of the angles in both triangles (you cannot use both ASA and AAS at the same time to say that two triangles are congruent).
On your homework or quiz I can almost guarantee you’ll see a congruence problem just like this. Where you’re given parallel lines and some transversal. What you have to remember is that alternate interior angles will be congruent.
So let’s start if I’m looking at these two triangles there’s only one thing that I know and that’s that line segment ZV is congruent to line segment XY but that’s not enough to say that those two triangles are congruent. So let’s do the whole problem solving.
Well I see right here that we have vertical angles which are always congruent. So I’m going to mark angle W as congruent to angle W, or to be technical this is angle XWY and this is angle ZWV. But we need one more piece of information and that’s going to use our transversal ZY. Alternate interior angles will be congruent and notice that I’m using two marks here instead of one because these two angles are congruent to each other but not necessarily to angle W.
So our congruent short-cut here I’m going to say these two triangles are congruent by angle-angle-side. Notice the side is not in between the two angles so it’s not included. That’s why we say angle-angle-side. The second step is to say what are the corresponding vertices?
Well angle W which is right here corresponds to angle W in the lower triangle so I’m going to write that first. Next angle Y, which comes second in this listing, corresponds to Z so I’m going to write angle Z second.
And last angle X corresponds to angle V so I’m going to write that third, so back up what did we do? We found angles that are congruent so we could use the short-cut, angle-angle-side, and we made sure our vertices corresponded to each other.