Question

In the given figure, regular pentagons are drawn on each side of an equilateral triangle. The measure each angle between the pentagons is

• A. ${84}^{\circ }$
• B. ${94}^{\circ }$
• C. ${54}^{\circ }$
• D. ${64}^{\circ }$

Answer 1

The correct option is A

${84}^{\circ }$

For a regular pentagon n=5

$\therefore$ Each exterior angle = $\frac{{360}^{\circ }}{5}$

$={72}^{\circ }$

Now, exterior angle + Interior angle = ${180}^{\circ }$

$\Rightarrow e+i={180}^{\circ }$

$\Rightarrow {72}^{\circ }+i={180}^{\circ }$

$i={180}^{\circ }-{72}^{\circ }$

$i={108}^{\circ }$

Also, each angle of an equilateral triangle = ${60}^{\circ }$

Now, Angle about a point = ${360}^{\circ }$

${108}^{\circ }+{60}^{\circ }+{108}^{\circ }+x={360}^{\circ }$

${276}^{\circ }+x^{\circ }={360}^{\circ }$

$x^{\circ }={360}^{\circ }-{276}^{\circ }$

$x^{\circ }={84}^{\circ }$

$\therefore$ Each angle between the pentagons is ${84}^{\circ }$.