The document NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.

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**EXERCISE 3.1 ****Question 1.** **Given here are some figures.**

1.

2.

3.

4.

5.

6.

7.

8.**Classify each of them on the basis of the following:**

(a) Simple curve (b) Simple closed curve (c) Polygon

(d) Convex polygon (e) Concave **polygon ****Solution:**

(a) Simple curves are: (1), (2), (5), (6) and (7).

(b) Simple closed curves are: (1), (2), (5), (6) and (7).

(c) Polygons are: (1), (2) and (4).

(d) Convex polygon is: (2).

(e) Convex polygons are (1) and (4).**Question 2. ****How many diagonals does each of the following have? **

(a) A convex quadrilateral

(b) A regular hexagon **(c) A triangle****Solution:****Note:** Number of diagonals in a polygon of n-sides =

(a) In quadrilateral, number of sides (n) = 4

∴ Number of diagonals

(b) In a regular hexagon, number of sides (n) = 6

∴ Number of diagonals

(c) In a triangle, number of sides (n) = 3

∴ Number of diagonals **Question 3.** **What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)****Solution: **The sum of measures of angles of a convex quadrilateral = 360° Yes, this property holds, even if the quadrilateral is not convex.**Question ****4. ****Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that).**

Figure | ||||

Side | 3 | 4 | 5 | 6 |

Angle sum | 180° | 2 x 180° = (4 – 2) x 180° | 3 x 180° = (5 – 2) x 180° | 4 x 180° = (6 – 2) x 180° |

**What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b) 8 (c) 10 (d) n****Solution: **From the above table, we conclude that sum of the interior angles of polygon with n-sides = (n – 2) x 180°

(a) When n = 7

Substituting n = 7 in the above formula, we have

Sum of interior angles of a polygon of 7 sides (i.e. when n = 7)

= (n – 2) x 180° = (7 – 2) x 180°

= 5 x 180° = 900°

(b) When n = 8

Substituting n = 8 in the above formula, we have

Sum of interior angles of a polygon having 8 sides

= (n – 2) x 180° = (8 – 2) x 180°

= 6 x 180° = 1080°

(c) When n = 10

Substituting n = 10 in the above formula, we have

Sum of interior angles of a polygon having 10 sides

= (n – 2) x 180°

= (10 – 2) x 180°

= 8 x 180° = 1440°

(d) When n = n

The sum of interior angles of a polygon having n-sides = (n – 2) x 180°

**Question 5.**** What is a regular polygon?**

State the name of a regular polygon of

(i) 3 sides

(ii) 4 sides **(iii) 6 sides**

**Solution:** A polygon is said to be a regular polygon if

(a) The measures of its interior angles are equal, and

(b) The lengths of its sides are equal.

The name of a regular polygon having

(i) 3 sides is ‘equilateral-triangle’

(ii) 4 sides is ‘square’

(iii) 6 sides is ‘regular hexagon’**Question 6. ****Find the angle measure x in the following figures.****Solution: (a)** ∵ The sum of interior angles of a quadrilateral = 360°

∴ x + 120° + 130° + 50° = 360°

or x + 300° = 360°

or x = 360° – 300° = 60°

(b) ∵ The sum of interior angles of a quadrilateral = 360°

∴ x + 60° + 70° + 90° = 360°

or x + 220° = 360°

or x = 360° – 220° = 140°

(c) Interior angles are: 30°, x°, x°, (180° – 70°)

and (180° – 60°),i.e. 30°, x°, x°, 110° and 120°

The given figure is a pentagon.

∵ Sum of interior angles of a pentagon = 540°

∴ 30° + x + x + 110° + 120° = 540°

or

2x + 260° = 540°

or

(d) It is a regular pentagon.

Sum of all interior angles of a regular pentagon = 540°.

∴ Its each angle is equal.

∴ x + x + x + x + x = 540°

or

5x = 540°

or

x = 540° ÷ 5 = 108°**Question 7.****(a) Find x + y + z. ****(b) Find x + y + z + w.****Solution:**

(a) ∵ x + 90° = 180° [Linear pair]

∴ x = 180° – 90° = 90°

y = 30° + 90° = 120° [∵ Sum of interior opposite angles = exterior angle]

z = 180° – 30° = 150°

Now, x + y + z = 90° + 120° + 150° = 360°

(b) ∵ The sum of interior angles of a quadrilateral = 360°

∴ ∠1 + 120° + 80° + 60° = 360°

or ∠1 + 260° = 360°

or ∠1 = 360° – 260° = 100°

Now, x + 120° = 180° [Linear pair]

∴ x = 180° – 120° = 60°

y + 80° = 180° [Linear pair]

∴ y = 180° – 80° = 100°

z + 60 = 180° [Linear pair]

∴ z = 180° – 60° = 120°

w + 100 = 180°" [Linear pair]

∴ w = 180° – 100° = 80°

Thus,

x + y + z + w = 60° + 100° + 120° + 80° = 360°

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