Question

The sum of the angles in the four segments exterior to a cyclic quadrilateral is

• A.
  ${360}^{\circ }$
• B.
  ${270}^{\circ }$
• C.
  ${540}^{\circ }$
• D.
  ${180}^{\circ }$

Answer 1

The correct option is C

${540}^{\circ }$
Let ABCD be a cyclic quadrilateral and $\angle P,\angle Q,\angle Rand\angle S$ be the angles in the four external segments.

Join SB and SC.

Since APBS is a cyclic quadrilateral.




$\therefore \angle ASB+\angle P={180}^{\circ }\dots ..\left(i\right)$

Similarly, BQCS is a cyclic quadrilateral.

$\therefore \angle CSB+\angle Q={180}^{\circ }\dots ..\left(ii\right)\text{and, CRDS is a cyclic quadrilateral.}\therefore \angle CSD+\angle R={180}^{\circ }\dots ..\left(iii\right)\text{Adding (i), (ii) and (iii), we get}(\angle ASB+\angle P)+(\angle CSB+\angle Q)+(\angle CSD+\angle R)=3\times {180}^{\circ }\Rightarrow \angle P+\angle Q+\angle R+(\angle ASB+\angle CSB+\angle CSD)={540}^{\circ }\Rightarrow \angle P+\angle Q+\angle R+\angle S={540}^{\circ }$

Thus, the sum of the angles in the four segments exterior to cyclic quadrilateral is 540º.


Hence, the correct answer is option (C).