Question

The alternate sides of any regular pentagon are produced to meet so as to form a star-shaped figure, as shown in the given figure.
Find the value of $\angle x+\angle y+\angle z+\angle t+\angle u$.

• A. ${90}^o$
• B. ${180}^0$
• C. ${270}^o$
• D. ${360}^o$

Answer 1

The correct option is B ${180}^0$

Each interior angle of regular pentagon $=\frac{(5-2)\times {180}^{\circ }}{5}={108}^{\circ }$
$\therefore$ Ext. $\angle BAK=Ext.\angle ABK={180}^{\circ }-{108}^{\circ }={72}^{\circ }$
$\therefore$ In $\Delta ABK,x={180}^{\circ }-({72}^{\circ }+{72}^{\circ })$
$={180}^{\circ }-{144}^{\circ }={36}^{\circ }$
Similarly, we can show $u=t=y=z={36}^{\circ }$
$\therefore \angle x+\angle y+\angle z+\angle t+\angle u={36}^{\circ }\times 5={180}^{\circ }$