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°.
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.