Learning Task 2 Solve the following problems. (Show your complete solutions.) 1.) What is the sum of the interior angles of a po

Question

Learning Task 2
Solve the following problems. (Show your complete solutions.)
1.) What is the sum of the interior angles of a polygon having 8 sides?
2.) Find the measure of the exterior angle of a regular decagon.
3.) Find the number of diagonals of a nonagon.
4.) What is the measure of each angle of a regular nonagon?
5.) if the sum of the measures of the angles of a regular polygon is 900, then
what is the number of sides?

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Trúc Chi 4 years 2021-09-05T05:43:03+00:00 1 Answers 161 views 0

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  1. Dau
    0
    2021-09-05T05:44:09+00:00
    Reply

    Answer:

    1. 1080°

    2. 36°

    3. 27 diagonals

    4. 140°

    5. 7 sides

    Step-by-step explanation:

    1. The sum S of the interior angles of a polygon is given by;

    S = (n-2) x 180°

    Where;

    n = number of sides of the polygon.

    Therefore, to get the sum of the interior angles of a polygon having 8 sides, we substitute n = 8 into equation (i) as follows;

    S = (8 – 2) x 180°

    S = 6 x 180°

    S = 1080°

    Therefore, the sum of the interior angles of a polygon having 8 sides is 1080°

    2. The sum of interior angles of a polygon is given by;

    S = (n – 2) x 180°

    For a polygon which is a regular decagon where the number of sides n = 10, the sum is;

    S = (10 – 2) x 180°

    S = 8 x 180°

    S = 1440°

    Since the decagon is regular, then each of its interior angles is given by;

    1440° / 10

    => 144°

    Now, since we know that the sum of the interior angle and exterior angle will give 180°, then;

    The exterior angle of a regular decagon is;

    180° – 144° = 36°

    Therefore, the exterior angle of a regular decagon is 36°

    3. The number of diagonals N of a polygon with n sides is given by;

    N = \frac{1}{2} n(n-3)

    So, for a nonagon which has 9 sides (i.e n = 9), the number of diagonals is;

    N = \frac{1}{2}*9(9-3)

    N = \frac{1}{2}*9(6)

    N = 9(3)

    N = 27

    Therefore, a nonagon has 27 diagonals.

    4. The sum of interior angles of a polygon is given by;

    S = (n – 2) x 180°

    For a polygon which is a nonagon where the number of sides n = 9, the sum is;

    S = (9 – 2) x 180°

    S = 7 x 180°

    S = 1260°

    Since the nonagon is regular, then each of its interior angles is given by;

    1260° / 9

    => 140°

    Therefore, the measure of each angle (interior angle) of a regular nonagon is 140°

    5. The sum of the interior angles of a polygon is given by;

    S = (n – 2) x 180°

    So if the sum is 900, then to get the number of sides of the polygon, we substitute S = 900 into the equation;

    900° = (n – 2) x 180°

    Divide through by 180°

    5 = (n – 2)

    Solve for n;

    n = 5 + 2

    n = 7

    Therefore, the regular polygon has 7 sides.

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