Here we will talk about the to organize of amount of the interiorangles of an n-sided polygon and some related instance problems.

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The sum of the inner angles the a polygon the n sides isequal to (2n - 4) appropriate angles.

Given: permit PQRS .... Z it is in a polygon of n sides.

To prove: ∠P + ∠Q + ∠R + ∠S + ..... + ∠Z = (2n – 4) 90°.


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Statement

Reason

1. Together the polygon has n sides, n triangles room formed,namely, ∆OPQ, ∆QR, ...., ∆OZP.

1. On each side of the polygon one triangle has actually been drawn.

2. The amount of all the angles of the n triangle is 2n rightangles.

2. The amount of the angle of every triangle is 2 appropriate angles.

3. ∠P + ∠Q + ∠R + ..... + ∠Z + (sum of every anglesformed at O) = 2n appropriate angles.

3. From statement 2.

4. ∠P + ∠Q + ∠R + ..... + ∠Z + 4 ideal angles = 2n rightangles.

4. Amount of angles roughly the suggest O is 4 ideal angles.

5. ∠P + ∠Q + ∠R + ..... + ∠Z

           = 2n best angles - 4 best angles

           = (2n – 4) best angles

           =(2n – 4) 90°.        (Proved)

5. Indigenous statement 4.


Note:

1. In a consistent polygon of n sides, every angles space equal.

Therefore, each interior angle = (frac(2n - 4) × 90°n).

2. A quadrilateral is a polygon because that which n = 4.

Therefore, the amount of internal angles that a quadrilateral =(2 × 4 – 4) × 90° = 360°

 

Solved instances on finding the sum of the interior angles ofan n-sided polygon:

1. find the amount of the internal angles of a polygon of sevensides.

Solution:

Here, n = 7.

Sum the the internal angles = (2n – 4) × 90°

                                      = (2 × 7 - 4) × 90°

                                      = 900°

Therefore, the sum of the inner angles of a polygon is 900°.

2. amount of the interior angles the a polygon is 540°. Uncover thenumber of political parties of the polygon.

Solution:

Let the number of sides = n.

Therefore, (2n – 4) × 90° = 540°

⟹ 2n - 4 = (frac540°90°)

⟹ 2n - 4 = 6

⟹ 2n = 6 + 4

⟹ 2n = 10

⟹ n = (frac102)

⟹ n = 5

Therefore, the number of sides of the polygon is 5.

3. discover the measure of each interior angle of a regularoctagon.

Solution:

Here, n = 8.

The measure up of each inner angle = (frac(2n– 4) × 90°n)

                                                   = (frac(2 × 8 – 4) × 90°8)

                                                   = (frac(16 – 4) × 90°8)

                                                   = (frac12 × 90°8)

                                                   = 135°

Therefore, the measure up of each inner angle of a regularoctagon is 135°.

4.

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The proportion of the number of sides of two continuous polygonsis 3:4, and also the ratio of the sum of their internal angles is 2:3. Discover thenumber of sides of each polygon.