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Unformatted text preview: ase.8 In number 3, the length of the one given side a
was too short to even form a triangle; in number 4, the length of a was just long enough to form a
right triangle; in 5, a was long enough, but not too long, so that two triangles were possible; and
in number 6, side a was long enough to form a triangle but too long to swing back and form two.
These four cases exemplify all of the possibilities in the Angle-Side-Side case which are summarized
in the following theorem.
Theorem 11.3. Suppose (α, a) and (γ, c) are intended to be angle-side pairs in a triangle where
α, a and c are given. Let h = c sin(α)
• If a < h, then no triangle exists which satisﬁes the given criteria.
• If a = h, then γ = 90◦ so exactly one (right) triangle exists which satisﬁes the criteria.
• If h < a < c, then two distinct triangles exist which satisfy the given criteria.
• If a ≥ c, then γ is acute and exactly one triangle exists which satisﬁes the given criteria
Theorem 11.3 is proved on a case-by-case basis. If a &...
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