Question

$ABCDE$ is a regular pentagon. If $AD$ is joined, then

• A. $ABCD$ is a parallelogram
• B. $ABCD$ is a rhombus
• C. $ABCD$ is a cyclic
• D. $ABCD$ is not a quadrilateral

Answer 1

The correct option is C $ABCD$ is a cyclic

In $\Delta AED,$

$\angle E={108}^0$ (Interior angle of pentagon)

$\therefore \angle EAD+\angle EDA+LE={108}^0\angle EAD=\angle EDA(\because AE=ED)\therefore \angle EAD=LEDA={36}^0\angle ABC=LEDC={108}^0$

(Interior angle of pentagon)

$\angle EDC=\angle ADC+LEDA\therefore \angle ADC={108}^0-{36}^0={72}^0\angle ABC+\angle ADC=108+72={180}^0$

$\therefore \square ABCD$ is cyclic since opposite angles add to ${180}^0$.