Question

In the adjoining figure, ABCD is a cyclic quadrilateral. Then r + s is equal to

• A. ${180}^{\circ }$
• B. $2c$
• C. ${180}^{\circ }+2c$.
• D. ${180}^{\circ }-2c$.

Answer 1

The correct option is D

${180}^{\circ }-2c$.

$c=c_1(Vert.opp.\angle s)andb=c+s(Ext.\angle )d=c_1+r(Ext.\angle )$
But $b+d={180}^{\circ }$ (Opp. $\angle s$, cyclic quad.)
$\Rightarrow c+s+c_1+r={180}^{\circ }\Rightarrow r+s+2c={180}^{\circ }\Rightarrow r+s={180}^{\circ }-2c$.