Question

Find the sum of all the angles at the five vertices of the adjoining star.

Answer 1

Solve for the sum of all angles of the star

The vertices of the star are numbered in the figure

The sum of interior angles (pentagon formed inside) is given by $=(5–2)\times {180}^{\circ }={540}^{\circ }$

Here, each interior angle measures $\frac{540°}{5}={108}^{\circ }$

The supplementary angle will be ${180}^{\circ }–{108}^{\circ }={72}^{\circ }$

The angle at the vertex is ${180}^{\circ }–{72}^{\circ }–{72}^{\circ }={36}^{\circ }$

$\therefore$ Each angle at the vertices of the star measures $36$ degrees.

The sum of all the angles at five vertices of the star $=5\times {36}^{\circ }$

$={180}^{\circ }$

Hence, the sum of all the angles at the five vertices of the star is ${180}^{\circ }$