Voi nota cu:
[tex]\measuredangle X[/tex]
masura unghiului X.
In primul rand, deoarece cele 4 unghiuri sunt unghiuri in jurul unui punct, putem scrie ca:
[tex]\measuredangle AOB+\measuredangle BOC+\measuredangle COD+\measuredangle DOA=360^{\circ}[/tex]
Pentru a fi mai usor, voi nota masura unghiului AOB cu x.
[tex]3\cdot\measuredangle AOB=4\cdot\measuredangle COD\implies \measuredangle COD=3/4\cdot\measuredangle AOB =3/4\cdot x[/tex]
Scriem unghiul AOC ca:
AOC = AOB + BOC
[tex]\measuredangle AOC =\measuredangle AOB+\measuredangle BOC=x+\measuredangle BOC[/tex]
Scriem unghiul BOD ca:
BOD = BOC + COD
[tex]\measuredangle BOD = \measuredangle BOC + \measuredangle COD = \measuredangle BOC+ 3/4\cdot x[/tex]
Dar stim ca:
[tex]4\cdot\measuredangle AOC=5\cdot\measuredangle BOD[/tex]
adica:
[tex]4(x+\measuredangle BOC)=5(\measuredangle BOC+3/4\cdot x)\implies[/tex]
[tex]4\cdot x+4\cdot\measuredangle BOC=5\cdot\measuredangle BOC + 15/4\cdot x[/tex]
[tex]\measuredangle BOC = 4x-15/4\cdot x =1/4\cdot x[/tex]
Apoi:
[tex]\measuredangle DOA=\measuredangle AOD =7\cdot\measuredangle BOC=7/4\cdot x[/tex]
Introducem acum tot ce am obtinut in prima ecuatie:
[tex]x+1/4\cdot x+3/4\cdot x+7/4\cdot x=360^{\circ}\implies[/tex]
[tex]15/4\cdot x =360^{\circ}\implies x=4/15\cdot360^{\circ}=96^{\circ}[/tex]
Asadar:
[tex]\measuredangle AOB=x=96^{\circ}[/tex]
[tex]\measuredangle BOC=1/4\cdot x=24^{\circ}[/tex]
[tex]\measuredangle COD=3/4\cdot x=72^{\circ}[/tex]
[tex]\measuredangle DOA =7/4\cdot x =168^{\circ}[/tex]
