Question

Two lines AB and CD intersect at O. If $\angle AOC+\angle COB+\angle BOD=270°$, find the measures of $\angle AOC,\angle COB,\angle BOD$ and $\angle DOA.$

Answer 1

To find: and

Given: Two lines AB and CD intersects and ______(1)

We know that :

1. The sum of all the four interior angles is 360^o in a quadrilateral.
2. The sum of angles of a linear pair is 180°.
3. Vertically opposite angles are equal.

$\angle AOC+\angle COB+\angle BOD+\angle AOD=360°$(angle sum property of a quadrilateral) ______(2)

substituting equation (1) in (2)

$$\begin{gathered} 270°+\angle AOD=360° \\ \Rightarrow \angle AOD=360°-270° \\ \Rightarrow \angle AOD=90°\_\_\_\_\_\_\left(3\right) \end{gathered}$$

$\angle AOD+\angle BOD=180°$(linear pair) _________(4)

Substituting equation (3) in (4)

$$\begin{gathered} 90°+\angle BOD=180° \\ \Rightarrow \angle BOD=90° \end{gathered}$$

$\angle AOD=\angle BOD=90°$(vertically opposite angle)

$\angle BOD=\angle AOC=90°$(vertically opposite angle)

Hence the measures of the angles are$\angle AOC=\angle COB=\angle BOD=\angle AOD=90°$.