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what is the simplest way to explain how to determine whether the resolution of a triangle (finding all its sides and angles, given 3 of them) using the sine theorem gives one or two solutions and whether the solution found is correct? I have explained congruent triangles to my pupils (ASA, SAS, SSA, etc), but they can get the wrong answer if they change the order of calculating the angle (for instance, finding angle alpha before angle beta, which gives a seemingly correct answer unless you verify that the smaller angle corresponds to the smaller side, and so on). They could double check with the cosine theorem, but it seems a bit long. Thank you

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    $\begingroup$ SSA does not prove congruence. There's an ambiguous case. $\endgroup$ May 22 at 14:36

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Summarising this page: when solving $$\frac{\sin A}{a}=\frac{\sin \color{red}B}{b}$$ for angle $\color{red}B,$ and the triangle altitude that intersects with sides $a$ and $b$ has height $h,$ ambiguity arises precisely when

  • angle $A$ is acute  and
  • $h<a<b.$

enter image description here


Sue VanHattum's comment is pertinent:

I like this diagram, and I teach my students to draw the unknown side horizontally, so that their diagram will look like this. Then draw angle A, then side b. Then you can imagine side a swinging back and forth, and finding its two positions.

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    $\begingroup$ I like this diagram, and I teach my students to draw the unknown side horizontally, so that their diagram will look like this. Then draw angle A, then side b. Then you can imagine side a swinging back and forth, and finding its two positions. $\endgroup$
    – Sue VanHattum
    May 22 at 23:15

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