Seção II: Ângulos ⇒ Problema 112 - Linhas Retas e Ângulos-Vol. 1 Tópico resolvido

[petrasMOD]

Dado os ângulos consecutivos AOB, BOC e COD tal que:
[tex3]m\angle BOD = 80^o [/tex3] e a medida formada pelas bissetrizes dos ângulos AOB e COD é 90^o
Se [tex3]m\angle AOB + m\angle COD =80^o[/tex3] calcular [tex3]m\angle AOB[/tex3]

Resposta

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Gabarito: C) 50^o

[rcompany]

[tex3]R,S\text{ tais que }(OR)\text{ bissetriz de }\angle AOB\text{ e }(OS)\text{ bissetriz de }\angle COD\\
m\angle AOR=a,\,m\angle ROB=r,\,m\angle BOC=b,\,m\angle COS=c,\,m\angle SOD=s\\\\
\begin{array}{rl}
\left.\begin{array}{r}m\angle BOD = 80°\\
m\angle ROS=90°\\
m\angle AOB + m\angle COD =80°\\
(OR)\text{ bissetriz de} \angle AOB\\
(OS)\text{bissetriz de }\angle COD
\end{array}\right\}
&\implies
\left\{\begin{array}{l} b+c+s=80°\\ r+b+c=90°\\ a+r+c+s=80° \\a=r\\c=r\end{array}\right.\\
&\implies
\left\{\begin{array}{l} b+2c=80°\\ a+b+c=90°\\ a+c=40° \end{array}\right.\\
&\implies
\left\{\begin{array}{l} b+2c=80°\\ 2a+b+2c=130° \end{array}\right.\\
&\implies 2a=50°=\angle AOB
\end{array}[/tex3]