How Do You Know if a Triangle Is Ssa or Ass
Discover: This material will be included in a forthcoming (summer 2000) book with the tentative championship Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. This new book volition be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This fabric is in draft course and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the textile with students. Equally always comments are welcome and volition affect the final draft. Ship comments to dwh2@cornell.edu.
Chapter nine
SSS, ASS, SAA, and AAA
Things which coincide with one another are equal to one another.
— Euclid, Elements, Mutual Notion 4 [A: Euclid]
This chapter is a continuation of the triangle congruence properties studied in Affiliate 6.
Trouble nine.i. Side-Side-Side (SSS)
Are two triangles coinciding if the 2 triangles have congruent respective sides ?
Figure 9.1. SSS.
Suggestions
Commencement investigating SSS by making two triangles coincide every bit much as possible, and come across what happens. For example, in Effigy ix.2, if we line up one pair of corresponding sides of the triangles, nosotros have two dissimilar orientations for the other pairs of sides:
Figure 9.ii. Are these possible?
Of course, it is upward to you to decide if each of these orientations is actually possible, and to prove or disprove SSS. Once more, symmetry can be very useful here.
On a sphere, SSS doesn't work for all triangles. The counterexample in Effigy nine.3 shows that no matter how small the sides of the triangle are, SSS does not concord considering the three sides always make up one's mind two different triangles on a sphere. Thus, information technology is necessary to restrict the size of more than but the sides in order for SSS to hold on a sphere. Whatever argument you used for the plane should work on the sphere for suitably divers small triangles. Make sure you see what information technology is in your statement that doesn't piece of work for large triangles.
Figure 9.3 . A large triangle with small sides.
There are also other types of counterexamples to SSS on a sphere. Can y'all find them?
Trouble 9.2. Bending-Side-Side (Donkey)
a. Are ii triangles coinciding if an angle, an adjacent side, and the contrary side of ane triangle are coinciding to an angle, an next side, and the opposite side of the other?
Effigy nine.4. ASS.
Suggestions
Suppose you have two triangles with the above congruencies. We will call them ASS triangles. Nosotros would like to see if, in fact, the triangles are congruent. We can line upward the angle and the outset side, and we know the length of the 2d side (bc orb'c'), merely we don't know where the second and third sides will encounter. Run across Figure 9.five.
Figure 9.5. ASS is not truthful, in full general .
Here, the circle that has as its radius the second side of the triangle intersects the ray that goes from a along the anglea tob twice. So ASS doesn't work for all triangles on either the plane or a sphere or a hyperbolic airplane. Attempt this for yourself on these surfaces to see what happens. Can yous make Donkey work for an appropriately restricted form of triangles? On a sphere, also look at triangles with multiple correct angles, and, once more, define "small" triangles as necessary. Your definition of small triangle here may exist very dissimilar from your definitions in Problems half dozen.3 and 6.four.
There are numerous collections of triangles for which ASS is true. Explore. Come across what you lot find on all 3 surfaces.
b. Evidence that on the plane Ass hold for right triangles (where the Angle in Bending-Side-Side is right).
This issue is often called the Right-Leg-Hypotenuse Theorem (RLH), which tin can be expressed in the following manner:
On the plane, if the leg and hypotenuse of one right triangle are coinciding to the leg and hypotenuse of another right triangle, so the triangles are congruent .
What happens on a sphere and a hyperbolic plane?
At this point, you might conclude that RLH is truthful for modest triangles on a sphere. Only there are modest triangle counterexamples to RLH on spheres! The counterexample in Figure 9.6 will help y'all to encounter some ways in which spheres are intrinsically very different from the airplane. We can run into that the second leg of the triangle intersects the geodesic that contains the tertiary side an infinite number of times. So on a sphere there are small triangles which satisfy the conditions of RLH although they are not-congruent. What nearly on a hyperbolic airplane?
Figure 9.half dozen. Counterexample to RLH on a sphere.
Notwithstanding, if you look at your argument for RLH on the plane, you should be able to show that
On a sphere, RLH is valid for a triangle with all sides less than 1/4 of a dandy circumvolve.
RLH is also true for a much larger drove of triangles on a sphere. Can you find such a drove? What about on a hyperbolic airplane?
Problem 9.iii. Side-Angle-Angle (SAA)
Are 2 triangles congruent if 1 side, an adjacent bending, and the opposite angle of one triangle are congruent, respectively, to i side, an next angle, and the contrary angle of the other triangle?
Suggestions
As a full general strategy when investigating these problems, beginning by making the two triangles coincide as much as possible. You lot did this when investigating SSS and ASS. Let u.s.a. endeavor it as an initial step in our proof of SAA. Line up the first sides and the kickoff angles. Since we don't know the length of the 2nd side, we might end up with a motion-picture show like this:
Figure 9.7. Starting SAA.
The situation shown in Figure 9.7 may seem to you lot to exist impossible. You may exist asking yourself, "Tin this happen?" If your temptation is to argue that a and b cannot exist congruent angles and that it is not possible to construct such a figure, behold Effigy 9.eight.
Figure ix.8. A counterexample to SAA.
Yous may be suspicious of this example because information technology is not a counterexample on the plane. You may feel sure that it is the only counterexample to SAA on a sphere. In fact, nosotros can observe other counterexamples for SAA on a sphere.
With the assistance of parallel transport, yous can construct many counterexamples for SAA on a sphere. If you lot wait back to the first counterexample given for SAA, you can run across how this problem involves parallel ship, or similarly how it involves Euclid's Exterior Angle Theorem, which we looked at in Problem 8.i.
Can we brand restrictions such that SAA is true on a sphere? You should be able to reply this question by using the fuller agreement of parallel transport yous gained in Problems eight.ane and viii.two. Yous may be tempted to employ the result, the sum of the interior angles of a triangle is 180 ° , in order to evidence SAA on the plane. This result volition be proven after (Trouble 10.4) for the plane, merely nosotros saw in Problems vii.i and seven.2 that it does non hold on spheres and hyperbolic planes. Thus, nosotros encourage you to avoid using it and to employ the concept of parallel ship instead. This proposition stems from our desire to encounter what is common betwixt the airplane and the other two surfaces, as much as possible. In add-on, before we can prove that the sum of the angles of a triangle is 180°, nosotros will have to brand some additional assumptions on the plane which are not needed for SAA.
Trouble 9.4. Angle-Angle-Bending (AAA)
Are two triangles coinciding if their corresponding angles are congruent?
Figure ix.9. AAA.
Equally with the three previous problems, make the two AAA triangles coincide every bit much equally possible. Nosotros know that we tin line upwardly one of the angles, merely we don't know the lengths of either of the sides coming from this bending. So there are ii possibilities: (1) Both sides of i are longer than both sides of the other, as the instance in Figure 9.x shows on the plane.
Figure 9.10. Is this possible?
or (2) 1 side of the first triangle is longer than the corresponding side of the second triangle and vice versa, as the example in Figure nine.11 shows on a sphere.
Figure nine.11. Is this possible?
As with Problem 9.3, yous may call back that the example in Figure 9.11 cannot happen on either a plane, a sphere, or on a hyperbolic airplane. The possible being of a counterexample relies heavily on parallel transport — you can identify the parallel transports in each of the examples given. Try each counterexample on the plane, on a sphere, and on a hyperbolic airplane and see what happens. If these examples are not possible, explain why, and if they are possible, see if you can restrict the triangles sufficiently and so that AAA does hold.
Parallel send shows upwardly in AAA, similar to how it did in SAA, but here it happens simultaneously in two places. In this case, yous will recognize that parallel transport produces like triangles that are not necessarily congruent. Still, at that place are no like triangles on either a sphere or a hyperbolic aeroplane (equally you will run across after you finish AAA) and and so you certainly need a proof that shows why such a construction is possible and why the triangles are not coinciding. The construction may seem intuitively possible to you, just you lot should justify why it is a counterexample. Once more yous may demand properties of parallel lines from Problems viii.ane through viii.three. You lot may also need the property of parallel send on the plane stated in Problem 10.1 — you tin assume this holding now equally long as you are sure not to apply AAA when proving it later.
On a sphere or on a hyperbolic plane, is it possible to make the two parallel ship constructions shown in Figure ix.5 and thus get two non-congruent triangles? Try it and run across. Information technology is important that you brand such constructions and that you lot report them on a model of a sphere and of a hyperbolic plane.
Source: https://pi.math.cornell.edu/~dwh/books/eg99/Ch09/Ch09.html
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