I don't know how to prove or disprove the following problem: How to construct triangle if elements $a$, $b$, $\beta-\gamma$ are given? Is it constructible (if not, how to prove it)? Any help is welcome.
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I don't think that such a triangle can be constructed. A triangle construction needs 3 values known. These three values should be valid Congruence rule (such as SAS, SSS, ASA etc.).
Here, we are given 2 edges and difference of angles opposite to those sides. The two edges are 2 values and for third, we have no usable means to deduce. Just one relation between two angles is not enough.
with $a,\angle B,C$ you can construct a triangle.but the extra b may cause conflict unless you ignore it. or you can use $a,b , \angle C$ to construct a triangle.but you have to omit $\angle B$.