Question

Small triangle. Answer the following.

a. State a definition for small triangle that would eliminate a counterexample to ASA.

b. Explain why using this definition eliminates the possibility of a counterexample to ASA.

Answer 1

The Angle-Side-Angle (ASA) Rule states that -

If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

An included side is the side between the two given angles.

When considering ASA, both of the angles must be on the same side — the interior of the triangle. For example,

Let us look at a proof of ASA on the plane:

Figure ASA on the plane.

The planar argument for ASA does not work on spheres, cylinders, and cones because, in general, geodesics on these surfaces intersect in more than one point. But can you make the planar work on a hyperbolic plane? (You may want to modify the planar proof to use only reflections.)

As was the case for SAS, we must ask ourselves if we can find a class of small triangles on each of the different surfaces for which the above argument is valid. You should check if your previous definitions of small triangle are too weak, too strong, or just right to make ASA true on spheres, cylinders, and cones. It is also important to look at cases for which ASA does not hold. Just as with SAS, some interesting counterexamples arise.

In particular, try out the configuration in Figure 6.9 on a sphere. To see what happens you will need to try this on an actual sphere. If you extend the two sides to great circles, what happens? You may instinctively say that it is not possible for this to be a triangle, and on the plane most people would agree, but try it on a sphere and see what happens. Does it define a unique triangle? Remember that on a sphere two geodesics always intersect twice.

Figure 6.9. Possible counterexample to ASA.

Finally, notice that in our proof of ASA on the plane, we did not use the fact that the sum of the angles in a triangle is 180°. We avoided this for two reasons. For one thing, to use this "fact" we would have to prove it first. As we have already discussed in Chapter 4, this is both time consuming and unnecessary. We will prove it later (Problem 10.4). The other reason is that a proof using the fact that the angles sum to 180° will not work on a sphere or on a hyperbolic plane because there are at least some triangles on these surfaces whose angles sum to other than 180°. A common example is the triple-right triangle on a sphere, depicted in Figure 6.10, which you may have seen before. In hyperbolic triangles the sum of the angles appears to be less than 180°, see Figure 6.11 for an example.

Figure ]Triple-right triangle on a sphere.