Math Problem Statement

Find the missing values of angles of 'x' in each of the following polygons.

Solution

Let's solve the given question step by step for each polygon. The goal is to find the missing angle xx in each part:


(a) Triangle

In any triangle, the sum of the angles is 180180^\circ. The given angles are 3030^\circ and 9090^\circ. Thus: x+90+30=180x + 90^\circ + 30^\circ = 180^\circ x=180120x = 180^\circ - 120^\circ x=60x = 60^\circ


(b) Quadrilateral

In any quadrilateral, the sum of the angles is 360360^\circ. The given angles are 8080^\circ, 6060^\circ, and a right angle (9090^\circ). Thus: x+80+60+90=360x + 80^\circ + 60^\circ + 90^\circ = 360^\circ x=360(80+60+90)x = 360^\circ - (80^\circ + 60^\circ + 90^\circ) x=360230x = 360^\circ - 230^\circ x=130x = 130^\circ


(c) Pentagon

In any pentagon, the sum of the angles is (n2)180(n-2) \cdot 180^\circ, where n=5n = 5. Thus: Sum of angles=(52)180=540\text{Sum of angles} = (5-2) \cdot 180^\circ = 540^\circ All the other angles are 120120^\circ each, and the last angle is xx. Thus: 120+120+120+120+x=540120^\circ + 120^\circ + 120^\circ + 120^\circ + x = 540^\circ x=540480x = 540^\circ - 480^\circ x=60x = 60^\circ


(d) Triangle

In any triangle, the sum of the angles is 180180^\circ. The given angles are 6868^\circ and 120120^\circ. Thus: x+68+120=180x + 68^\circ + 120^\circ = 180^\circ x=180188x = 180^\circ - 188^\circ x=8x = -8^\circ (Might want to recheck on the input values -- double-check!)

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Math Problem Analysis

Mathematical Concepts

Geometry
Polygon angle sum
Triangle angle properties

Formulas

Sum of angles in a triangle = 180°
Sum of angles in a polygon = (n-2) × 180°

Theorems

Angle sum property of triangles
Angle sum property of polygons

Suitable Grade Level

Grades 6-8