In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles $ABC$ and $DEF$ such that $AB = DE$, $BC = EF$, and $\angle A = \angle D$, then we cannot deduce that $ABC$ and $DEF$ are congruent. However, there are a few special cases in which SSA "works". That is, suppose $ABC$ is a triangle. Let $AB = x$, $BC = y$, and $\angle A = \theta$. For some values of $x$, $y$, and $\theta$, we can uniquely determine the third side, $AC$. (a) Use the Law of Cosines to derive a quadratic equation in $AC$. (b) Use the quadratic polynomial you found in part (a) in order to find conditions on $x, y,$ and $\theta$ which guarantee that the side $AC$ is uniquely determined.

 Jul 17, 2020

For the community: Please don't submit solutions to this problem. This is a homework problem for an online course that does not allow students to search for answers to specific problems outside of the school.


For the original poster: We realize that homework can be challenging. If you wish to receive help from the staff or other students, we encourage you to use the resources that the online classes provide. Please don't ask or search online for homework help. We understand that it's common in today's information age to look for resources online, and in some contexts, that's a great thing! However, it's against our Honor Code. You can ask for help on the message boards, and you can learn from the official solution after you submit your answer.

 Aug 3, 2021

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