Question

Two lines AB and CD intersect at a point O, such that ∠BOC + ∠AOD = 280°, as shown in the figure. Find all the four angles.

Answer 1

We know that if two lines intersect, then the vertically-opposite angles are equal.
Let $\angle BOC=\angle AOD=x°$
Then,
$$\begin{gathered} x+x=280 \\ \Rightarrow 2x=280 \\ \Rightarrow x=140° \\ \therefore \angle BOC=\angle AOD=140° \end{gathered}$$
Also, let $\angle AOC=\angle BOD=y°$
We know that the sum of all angles around a point is $360°$.
$$\begin{gathered} \therefore \angle AOC+\angle BOC+\angle BOD+\angle AOD=360° \\ \Rightarrow y+140+y+140=360° \\ \Rightarrow 2y=80° \\ \Rightarrow y=40° \end{gathered}$$
Hence, $\angle AOC=\angle BOD=40°$
$\therefore \angle BOC=\angle AOD=140°\text{and}\angle AOC=\angle BOD=40°$