10. Five identical right-angled triangles can be arranged so that their larger acute angles touch to form the star shown in the diagram. It is also possible to form a different star by arranging more of these triangles so that their smaller acute angles touch.
How many triangles are needed to form the second star?

A 10
B 12
C 18
D 20
E 24
SolUtion
D

Since the five identical triangles meet at a point, the size of the larger acute angle in each triangle is $360^{\circ} \div 5=72^{\circ}$. Therefore the smaller acute angle in each triangle is $180^{\circ}-90^{\circ}-72^{\circ}=18^{\circ}$. Hence, since the second star is formed using the triangles whose smaller acute angles touch, the number of triangles needed to form the second star is $360 \div 18=20$.