Bài giải:

Ta có: $\widehat{{{O}_{1}}}+\widehat{{{O}_{3}}}=\frac{1}{2}(\widehat{{{O}_{2}}}+\widehat{{{O}_{4}}})$            (theo đề bài)

Mà: $\widehat{{{O}_{1}}}=\widehat{{{O}_{3}}}$ ; $\widehat{{{O}_{2}}}=\widehat{{{O}_{4}}}$ (hai góc đối đỉnh)

Suy ra: $2.\widehat{{{O}_{1}}}=\frac{1}{2}.2.\widehat{{{O}_{4}}}$

$\Rightarrow $ $\widehat{{{O}_{1}}}=\frac{1}{2}\widehat{{{O}_{4}}}$

Mà: $\widehat{{{O}_{1}}}+\widehat{{{O}_{4}}}={{180}^{0}}$ (tổng hai góc kề bù)

$\Rightarrow $$\frac{1}{2}\widehat{{{O}_{4}}}+\widehat{{{O}_{4}}}={{180}^{0}}=\frac{3}{2}.\widehat{{{O}_{4}}}$ 

$\Rightarrow $$\widehat{{{O}_{4}}}={{120}^{0}}$ 

Vậy $\widehat{{{O}_{4}}}={{120}^{0}}$