Question

(a) For the figure below, calculate the missing angle x.

N= 125°
E=97⁰
P=115⁰
T=x
A = 85°

(b) A regular polygon has an exterior angle measure of (x+3) and an interior angle measure of (13x - 33).
Find the measure of each angle.
Interior angle =
Exterior =

Answer 1

The sum of angles in a polygon and the sum of an interior and exterior angle of a polygon indicates;

(a) x = 118°

(b) Interior angle = 162°

Exterior angle = 18°

What is the sum of an exterior and interior angle of a polygon; An exterior angle of a polygon and the adjacent interior angle of the polygon are supplementary, therefore, the sum of an exterior angle and the adjacent interior angle is 180°

(a) The number of vertices indicated by the angles ∠N, ∠E, ∠P, ∠T, and ∠A, are 5 vertices, which indicates that the possible figure is a pentagon

Please find attached the drawing of the possible figure NEPTA in the question created with MS Word;

Whereby the figure is a pentagon, we get;

The sum of the interior angles in a pentagon = 540°

Therefore; ∠N + ∠E + ∠P + ∠T + ∠A = 540°

125 + 97 + 115 + x + 85 = 540

x = 540 - (125 + 97 + 115 + 85)

540 - (125 + 97 + 115 + 85) = 118

x = 118°

(b) The interior and exterior angle of a polygon are supplementary, therefore;

The sum of the interior and exterior angle of the polygon is 180°

The measure of an exterior angle of the polygon = (x + 3)°

The measure of an interior angle of the polygon = (13·x - 33)°

(x + 3) + (13·x - 33) = 180

14·x - 30 = 180

x = (180 + 30)/14

(180 + 30)/14 = 15

x = 15

The measure of the exterior angle, x + 3 is; 15 + 3 = 18°

The measure of the interior angle, (13·x - 33) is; (13 × 15 - 33) = 162°