Question

In fig. 1.54 $\angle COD$ = 90$°$ $\angle$BOE = 72$°$ and $\angle$AOB is a straight angle. Find the measures of the following angles.
$\angle$AOC, $\angle$BOC, $\angle$AOE

Answer 1

Given:
$$\begin{gathered} \angle \mathrm{COD}=90° \\ \angle \mathrm{BOE}=72° \\ \angle \mathrm{AOB}\mathrm{is}\mathrm{a}\mathrm{straight}\mathrm{angle}. \end{gathered}$$

$$\begin{gathered} \angle \mathrm{AOC}=\angle \mathrm{BOE}(\mathrm{Vertically}\mathrm{opposite}\mathrm{angles}) \\ \Rightarrow \angle \mathrm{AOC}=72° \end{gathered}$$

$$\begin{gathered} \mathrm{Here},\angle \mathrm{AOC}+\angle \mathrm{COD}+\angle \mathrm{BOD}=180°(\angle \mathrm{AOB}\mathrm{is}\mathrm{a}\mathrm{straight}\mathrm{angle}) \\ \Rightarrow 72°+90°+\angle \mathrm{BOD}=180° \\ \Rightarrow 162°+\angle \mathrm{BOD}=180° \\ \Rightarrow \angle \mathrm{BOD}=18° \\ \mathrm{Now},\angle \mathrm{BOC}=\angle \mathrm{BOD}+\angle \mathrm{COD} \\ =18°+90° \\ =108° \end{gathered}$$

$$\begin{gathered} \mathrm{Thus},\angle \mathrm{AOE}=\angle \mathrm{BOC}(\mathrm{Vertically}\mathrm{opposite}\mathrm{angles}) \\ \mathrm{i}.\mathrm{e}.,\angle \mathrm{AOE}=108° \end{gathered}$$