Geometry / Mr. Hansen 10/14/2004

Name: _________________________

Notes and Hints for §3.2

SSS: If we can show that all 3 sides of one D are @ to all 3 corresponding sides of another D, then the triangles must be @. Warning: When stating the congruence, you must list the vertices in the proper order. For example, in #7 on p.121, it would be incorrect to say DTVA @ DAWX by SSS, since vertex T does not correspond to vertex A in the other triangle. You have to match up T with X, V with W, and A with A in order to satisfy the givens. Correct conclusion: DTVA @ DXWA.

ASA: If we can show that 2 angles of one D, as well as the side that is between them, are @ to the corresponding 2 angles and side between them in a second D, then the triangles must be @. The same warning applies as above.

SAS: If we can show that 2 sides of one D, as well as the angle that is between them, are @ to the corresponding 2 sides and angle between them in a second D, then the triangles must be @. The same warning applies as above, but there is a second warning as well. This is a big one, so listen up:

WARNING: Although SAS is a valid method to show two triangles @, you must be super-careful that the angle is between the two sides. If the angle is not between the two sides, we say that we have an “SSA” situation instead of an “SAS” situation, and SSA is not valid in general. (You can remember this by noting that SSA spells a bad word, backwards.) In other words, triangles can sometimes satisfy SSA without being @.

Question for advanced students: If SSA is invalid, do we also have to be super-careful when applying ASA to make sure that the side is between the two angles that are known to be congruent to corresponding angles in the other triangle?
Answer: No. As we will learn later in the course, you can mangle ASA into AAS without causing a fallacy in reasoning. However, your textbook authors have taken care to avoid any problems that require AAS until such time as we can prove that AAS is indeed valid. (Basically, we must first prove that the angles of a triangle always add up to 180°. Once we know that, it is easy to prove AAS from ASA. But don’t worry about any of that for the moment, OK?)

1.(a) To use SAS, we would need to know segment HG @ segment OK; to use ASA, that ÐJ @ ÐM.
   (b,c) No hints provided; do these yourself.

2.(a) SAS
   (b) trick question (cannot prove @ because _________________)
   (c) another trick question (cannot prove @ because _________________)
   (d) ASA

3. Sketch of proof (you must do the whole thing): Prove segment BD @ segment BD (Reflex. Prop.); then apply SAS.

4. Sketch of proof (you must do the whole thing): Prove Ð3 @ Ð4 by applying something from Chapter 2; then apply ASA.

5. Sketch of proof (you must do the whole thing): Prove ÐROM and ÐROP are both rt. Ðs; then in another step prove that ÐROM @ ÐROP; then prove segment RO @ segment RO so that you can apply SAS.

6. Sketch of proof (you must do the whole thing): First, label ÐTSV as Ð1, ÐBSV as Ð2, ÐTVS as Ð3, and ÐBVS as Ð4. This simplifies things greatly. Then, use the givens to prove Ð1 @ Ð2, plus Ð3 @ Ð4. Then, use a certain property to prove that segment VS is @ to itself. Finally, apply ASA.

10. Write an equation involving the given perimeter value. After solving for x, check to see if all corresponding sides are @.

11. Oral preparation: “I would show ÐN @ ÐS since comps. of @ Ðs are comp. Then, I would use the remaining given so that I could apply ASA.”

12-16. Please mark your diagrams with penciled tick marks and angle marks so that you can give your oral preparation when called upon.