9, 13, 14, 15, and 19 have me confused

9514 1404 393

  9.  not possible

  13, 14, 19.  AAS — 13: WYX≅AYZ; 14: CFD≅CEB; 19: DEH≅GEF

  15, 16.  not possible

Step-by-step explanation:

9. You're looking for three sides, or two sides flanking a congruent angle. Here, the angle is not between the two sides, so the SAS theorem does not apply. Proving congruence is not possible.

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13, 14, 19. You have two adjacent angles, followed by a side, so the AAS theorem will prove congruence.  

When you write the congruence statement, you need to make sure the nodes are named in corresponding order. It can be useful to name the nodes in the same order as the elements you claim for congruence. For these, we name the two angle nodes first, then the other end of the side.

13: WYX≅AYZ; 14: CFD≅CEB; 19: DEH≅GEF

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15, 16. With only one angle and one side, it is not possible to prove congruence.

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Additional comment

With these theorems, you are looking for angles or sides that are adjacent (next to each other). In figure 9, there are two sides and an angle, so you might be tempted to claim SAS congruence. The name SAS means the angle must be between the two sides. In that figure, the two sides are adjacent, and the angle is adjacent to only one of the sides (not between them). Hence SAS does not apply.

(You will find later that two sides and a non-adjacent angle will define a unique triangle if, and only if, the angle is opposite the longest of the two given sides. In figures like these, no side lengths are given, and you are not allowed to rely on appearances.)

over the argument its not good to fight.