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Unformatted text preview: lt; h, then a < c sin(α). If a triangle were
to exist, the Law of Sines would have sin(γ ) = sin(α) so that sin(γ ) = c sin(α) > a = 1, which is
impossible. In the ﬁgure below, we see geometrically why this is the case. c a c
h = c sin(α) α a < h, no triangle a = h = c sin(α)
α a = h, γ = 90◦ Simply put, if a < h the side a is too short to connect to form a triangle. This means if a ≥ h,
we are always guaranteed to have at least one triangle, and the remaining parts of the theorem
tell us what kind and how many triangles to expect in each case. If a = h, then a = c sin(α) and
7 If this sounds familiar, it should. From high school Geometry, we know there are four congruence conditions for
triangles: Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), Side-Angle-Side (SAS) and Side-Side-Side (SSS). If we
are given information about a triangle that meets one of these four criteria, then we are guaranteed that exactly one
triangle exists which satisﬁes the given criteria.
In more reputable books, this is called the ‘Side-Side-Angle’ or S...
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