Question

The sides of a regular 9-sided polygon have been
extended to make a star, as shown below.
Calculate the size of angle x.
Give your answer in degrees (°).
x

Answer 1

Answer:

x = 100°

Step-by-step explanation:

As the given shape is a regular polygon, the triangles created by extending the sides are isosceles triangles.

To calculate the base angles of the isosceles triangle, find the interior angle of the regular polygon:

$\textsf{Interior angle of a regular polygon} = \dfrac{180^{\circ}(n-2)}{n}$

$\textsf{(where }n \textsf{ is the number of sides)}$

Therefore:

$\textsf{Interior angle of a regular nonagon} = \dfrac{180^{\circ}(9-2)}{9}=140^{\circ}$

As angles on a straight line sum to 180°, the base angle of the isosceles triangle is:

= 180° - interior angle

= 180° - 140°

= 40°

Interior angles of a triangle sum to 180°.

⇒ 2 base angles + x = 180°

⇒ 2 × 40° + x = 180°

⇒ 80° + x = 180°

⇒ x = 180° - 80°

⇒ x = 100°