Ta có:

 \(\widehat{xOz}=4\widehat{yOz}\)

\(\widehat{xOz}=2\widehat{zOt}\)

⇒\(2\widehat{yOz}=\widehat{zOt}\)

⇒\(\widehat{yOt}=\widehat{yOz}+\widehat{zOt}=3\widehat{yOz}=90^o\)

⇒\(\widehat{yOz}=30^o\)

⇒\(\widehat{xOz}=120^o\)

⇒\(\widehat{xOy}=\widehat{xOz}+\widehat{zOy}=120^o+30^o=150^o\)

Vậy \(\widehat{xOy}=150^o\)

a) Ta có:

 \(\widehat{AOD}=\widehat{AOB}-\widehat{DOB}\)  (1)

\(\widehat{BOC}=\widehat{AOB}-\widehat{AOC}\)  (2)

mà \(\widehat{AOC}=\widehat{BOD}=90^o\)

⇒\(\widehat{AOD}=\widehat{BOC}\left(đpcm\right)\)

b) Từ (1), (2) và \(\widehat{DOC}=\widehat{AOB}-\widehat{AOD}-\widehat{BOC}\) ta có:

\(\widehat{DOC}=\widehat{AOB}-\widehat{AOB}+90^o-\widehat{AOB}+90^o=180^o-\widehat{AOB}\)

⇒\(\widehat{AOB}+\widehat{DOC}=180^o\left(đpcm\right)\)

c) Ta có

\(\widehat{AOx}=\dfrac{\widehat{AOD}}{2}=\dfrac{\widehat{AOB}}{2}-45^o\)

\(\widehat{BOy}=\dfrac{\widehat{BOC}}{2}=\dfrac{\widehat{AOB}}{2}-45^o\)

mà \(\widehat{xOy}=\widehat{AOB}-\widehat{AOx}-\widehat{BOy}\)

⇒\(\widehat{xOy}=\widehat{AOB}-\dfrac{\widehat{AOB}}{2}+45^o-\dfrac{\widehat{AOB}}{2}+45^o=90^o\)

⇒Ox \(\perp\) Oy \(\left(đpcm\right)\)

Gọi \(\widehat{AOD}\) và  \(\widehat{COB}\) là hai góc đối đỉnh 

     tia OE là tia phân giác của góc AOD

     tia OF là tia đối của tia OE

Ta có: \(\widehat{AOE}\) và \(\widehat{FOB}\) là hai góc đối đỉnh => \(\widehat{AOE}\) = \(\widehat{FOB}\)

             \(\widehat{EOD}\) và \(\widehat{COF}\) là hai góc đối đỉnh =>  \(\widehat{EOD}\) = \(\widehat{COF}\)

Mà \(\widehat{AOE}\) = \(\widehat{EOD}\)

=> ​​\(\widehat{FOB}\) = \(\widehat{COF}\)

=> OF là tia phân giác của góc COB 

Ta có: tia OD là tia phân giác của góc AOC => \(\widehat{AOD}\) = \(\widehat{DOC}\) = \(\dfrac{1}{2}\)\(\widehat{AOC}\)

          tia OE là tia phân giác của góc OCB =>\(\widehat{COE}\) = \(\widehat{EOB}\)= \(\dfrac{1}{2}\)\(\widehat{COB}\)

=>  \(\widehat{DOC}\) + \(\widehat{COE}\) =  \(\dfrac{1}{2}\)\(\widehat{AOC}\) +  \(\dfrac{1}{2}\)\(\widehat{COB}\)

=>  \(\widehat{DOC}\) + \(\widehat{COE}\) =  \(\dfrac{1}{2}\)( \(\widehat{AOC}\) + \(\widehat{COB}\))

Mà  \(\widehat{DOC}\) và  \(\widehat{COE}\) là hai góc kề bù

=>  \(\widehat{DOC}\) + \(\widehat{COE}\) =  \(\dfrac{1}{2}\) . 180^o =>  \(\widehat{DOC}\) + \(\widehat{COE}\) = 90^o  ( đpcm)    

Ta có: \(\widehat{O_1}\) + \(\widehat{O_2}\) + \(\widehat{O_3}\) + \(\widehat{O_4}\) = 360^o

             \(\widehat{O_1}\) + \(\widehat{O_2}\) + \(\widehat{O_3}\) = 325^o

=> \(\widehat{O_4}\) = 360^o - (  \(\widehat{O_1}\) + \(\widehat{O_2}\) + \(\widehat{O_3}\) )

=>  \(\widehat{O_4}\) = 360^o - 325^o

^=>  \(\widehat{O_4}\) = 35^o

Vậy  \(\widehat{O_4}\) = 35^o