DAV Class 8 Maths Chapter 11 Worksheet 1
Understanding Quadrilaterals Worksheet 1
1. Find the sum of interior angles of hexagon \(ABCDEF\) by dividing it into triangles.
Solution
Hexagon \( ABCDEF \) can be divided into 4 triangles: \( \color{magenta} \triangle ABC,\ \triangle ACD,\ \triangle ADE,\ \triangle AEF \)Sum of angles in a triangle \(=180^\circ \)
Sum of interior angles of the hexagon \( ABCDEF = 4 \times 180^\circ \implies \color{green}{720^\circ} \)
Answer \(\color{red}{720^\circ}\)
2. Find the sum of interior angles of a polygon of given number of sides by using formula \( (n - 2) \times 180^\circ \)
(i) 12 sides
Solution
Sum of interior angles of a polygon having 12 sides. \[ \begin{aligned} n &\implies 12 \\ \text{Sum of interior angles}&= \color{magenta} (n - 2) \times 180^\circ \\ &= (12 - 2) \times 180^\circ \\ &= 10 \times 180^\circ \\ &= 1800^\circ \end{aligned} \]Answer \(\color{red}{1800^\circ}\)
(ii) 9 sides
Solution
Sum of interior angles of a polygon having 9 sides. \[ \begin{aligned} n &\implies 9 \\ \text{Sum of interior angles}&= \color{magenta} (n - 2) \times 180^\circ \\ &= (9 - 2) \times 180^\circ \\ &= 7 \times 180^\circ \\ &= 1260^\circ \end{aligned} \]Answer \(\color{red}{1260^\circ}\)
(iii) 22 sides
Solution
Sum of interior angles of a polygon having 22 sides. \[ \begin{aligned} n &\implies 22 \\ \text{Sum of interior angles} &= \color{magenta} (n - 2) \times 180^\circ \\ &= (22 - 2) \times 180^\circ \\ &= 20 \times 180^\circ \\ &= 3600^\circ \end{aligned} \]Answer \(\color{red}{3600^\circ}\)
3. Find the measure of each angle of a regular octagon.
Solution
\[ \begin{aligned} \text{Regular octagon} &= 8 \text{ sides} \\ n &\implies 8 \end{aligned} \] Measure of each angle of a regular octagon\[ \begin{aligned} &= \color{magenta} \frac{(n - 2) \times 180^\circ}{n}\\[6pt] &= \frac{(8 - 2) \times 180^\circ}{8}\\[6pt] &= \frac{ \cancel{6}^3 \times \cancel{180^\circ}^{ \ 45^\circ}}{{\cancel{8}_{\cancel4}}_1}\\[6pt] &= 3 \times 45^\circ \\[6pt] &= 135^\circ \end{aligned} \]Answer Measure of each angle of a regular octagon \( = \color{red}{135^\circ}\)
4. Four angles of a pentagon are \(100^\circ, 175^\circ, 85^\circ \text{ and } 75^\circ \). Find the fifth angle.
Solution
\[ \begin{aligned} \text{Pentagon} &= 5 \text{ sides} \\ n &\implies 5 \end{aligned} \] Sum of all interior angles of a pentagon\[ \begin{aligned} &= \color{magenta} {(n - 2) \times 180^\circ}\\ &= (5 - 2) \times 180^\circ \\ &= 3 \times 180^\circ \\ &= 540^\circ \\ \\ \text{Let the 5th angle be } &= x \\ \text{Sum of all angles} &= 540^\circ \\ 100^\circ + 175^\circ + 85^\circ + 75^\circ + x &= 540^\circ \\ 435^\circ + x &= 540^\circ \\ x &= 540^\circ - 435^\circ \\ x &= 105^\circ \end{aligned} \]Answer Fifth angle of pentagon \( = \color{red}{105^\circ}\)
5. Find the measure of \(x\) in the given figure.
Solution
Sum of measures of exterior angles of any polygon \( = 360^\circ \) \[ \begin{aligned} 45^\circ + 100^\circ + 65^\circ + 40^\circ + x & = 360^\circ \\ 250^\circ + x & = 360^\circ \\ x & = 360^\circ - 250^\circ \\ x & = 110^\circ \\ \end{aligned} \]Answer \( \color{red}{x=110^\circ} \)