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
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). It is important to check if the angles and sides in question are correspondent in order to show congruence.
If there is not enough information, such as the congruence of certain angles or sides, the congruence of the triangles cannot be determined.
If the triangles are congruent, it is important to name the triangles so that the corresponding segments and angles are in the same position of the name. For example, if angle DAC is congruent to angle BCA, angle DCA is congruent to angle BAC, and the shared side AC is congruent to itself, then triangle ABC is congruent to triangle CDA by AAS.
When you are looking at two triangles that share a common side, the easiest way to determine if they are congruent is to redraw one of the triangles. So that's what I’m going to do here because I can’t tell just by looking at this if we have enough information.
So I’m going to redraw this triangle, DAC, down below and I’m going to make it look just like our lower triangle ABC. So which angle corresponds to A? Well angle C here has our 90 degree angle. Angle B corresponds to angle D because there are no markings on that angle, which means our last angle over here is angle A. So I’m going to write that with two markings.
So now the question is do we have enough information? I know that these two corresponding angles are congruent, I know that threes two angles are corresponding and congruent, but that’s not enough information. As I said in the beginning we have a shared side, AC must be congruent to itself; that’s called the reflexive property. So notice that we have angle, an included side and an angle that’s enough information to say that these two triangles must be congruent. So I’m going to write by angle-side-angle, which is our congruent short-cut.
The second step is to say well what corresponds to angle A? Angle A, because I’ve redrawn it it’s pretty easy to see that C corresponds to angle A. Angle B which comes second corresponds to angle D, and last angle C corresponds to angle A.
So the trick to this problem is to redraw this top triangle below so that you can compare the triangles pretty easily.
