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a)

Nếu góc $A$ nhọn thì ${{0}^{0}}<\widehat{A}<90{}^\circ \Rightarrow \cos \widehat{A}>0$

$\Leftrightarrow \dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}>0\Leftrightarrow {{b}^{2}}+{{c}^{2}}-{{a}^{2}}>0$

$\Leftrightarrow {{b}^{2}}+{{c}^{2}}>{{a}^{2}}$

b)

Nếu góc $A$ tù thì $90{}^\circ <\widehat{A}<180{}^\circ \Rightarrow \cos \widehat{A}<0$

$\Leftrightarrow \dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}<0\Leftrightarrow {{b}^{2}}+{{c}^{2}}-{{a}^{2}}<0$

$\Leftrightarrow {{b}^{2}}+{{c}^{2}}<{{a}^{2}}$

c)

Nếu góc $A$ vuông thì $\widehat{A}=90{}^\circ \Rightarrow \cos \widehat{A}=0$

$\Leftrightarrow \dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}=0\Leftrightarrow {{b}^{2}}+{{c}^{2}}-{{a}^{2}}=0$

$\Leftrightarrow {{b}^{2}}+{{c}^{2}}={{a}^{2}}$