Question

If each interior angle of a regular polygon is 144°. The polygon has: * a. 24 sides b. 10 sides c. 12 sides d. 20 sides​

Answer 1

Step-by-step explanation:

Each interior angle in a polygon of 'n' sides : 180° - (360/n)°

Equation formed :

180° - (360/n)° = 144° (n = number of sides)

360/n = 36

36n = 360

n = 10

Hence, the above polygon has 10 sides.

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Answer 2

$\large\underline{\text{Required Knowledge}}$

$\red{\bigstar}$Angle sum of a triangle.

The angles of a triangle sum to $180^{\circ}$.

$\red{\bigstar}$The number of diagonals.

In a regular polygon of $n$ sides, we can draw $(n-3)$ diagonals from a point since we cannot connect adjacent sides or itself to get a diagonal. Also, inside is $(n-2)$ triangles.

$\red{\bigstar}$Property of regular polygons.

Every single angle is equal in regular polygons.

$\large\underline{\text{Solution}}$

Since there are $(n-2)$ triangles, the angle sum of all the triangles is $180^{\circ}\times(n-2)$.

Then since the regular polygon has $n$ equal angles, one angle will be $\dfrac{180^{\circ}\times(n-2)}{n}$.

Now we get the equation.

$\implies\dfrac{180^{\circ}\times(n-2)}{n}=144^{\circ}$

$\implies180^{\circ}\times n-360^{\circ}=144^{\circ}\times n$

$\implies(180^{\circ}-144^{\circ})\times n=360^{\circ}$

$\implies36^{\circ}\times n=360^{\circ}$

$\implies n=10$

Hence, the shape is a regular 10-sided polygon.