## Exercise 3.1 Page: 41

**1. Given here are some figures.**

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

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

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

Solution:

a) Simple curve: 1, 2, 5, 6 and 7

b) Simple closed curve: 1, 2, 5, 6 and 7

c) Polygon: 1 and 2

d) Convex polygon: 2

e) Concave polygon: 1

**2. How many diagonals does each of the following have?**

**a) A convex quadrilateral (b) A regular hexagon (c) A triangle**

Solution:

a) A convex quadrilateral: 2.

Join BC, Such that it divides ABCD into two triangles ΔABC and ΔBCD. In ΔABC,

∠1 + ∠2 + ∠3 = 180° (angle sum property of triangle)

In ΔBCD,

∠4 + ∠5 + ∠6 = 180° (angle sum property of triangle)

∴, ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180°

⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°

⇒ ∠A + ∠B + ∠C + ∠D = 360°

Thus, this property hold if the quadrilateral is not convex.

**4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)**

**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:

The angle sum of a polygon having side n = (n-2)×180°

a) 7

Here, n = 7

Thus, angle sum = (7-2)×180° = 5×180° = 900°

b) 8

Here, n = 8

Thus, angle sum = (8-2)×180° = 6×180° = 1080°

c) 10

Here, n = 10

Thus, angle sum = (10-2)×180° = 8×180° = 1440°

d) n

Here, n = n

Thus, angle sum = (n-2)×180°

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

**State the name of a regular polygon of**

**(i) 3 sides (ii) 4 sides (iii) 6 sides **Solution:

Regular polygon: A polygon having sides of equal length and angles of equal measures is called regular polygon. i.e., A regular polygon is both equilateral and equiangular.

(i) A regular polygon of 3 sides is called equilateral triangle.

(ii) A regular polygon of 4 sides is called square.

(iii) A regular polygon of 6 sides is called regular hexagon.

**6. Find the angle measure x in the following figures.**

Solution:

a) The figure is having 4 sides. Hence, it is a quadrilateral. Sum of angles of the quadrilateral = 360°

⇒ 50° + 130° + 120° + x = 360°

⇒ 300° + x = 360°

⇒ x = 360° – 300° = 60°

b) The figure is having 4 sides. Hence, it is a quadrilateral. Also, one side is perpendicular forming right angle.

Sum of angles of the quadrilateral = 360°

⇒ 90° + 70° + 60° + x = 360°

⇒ 220° + x = 360°

⇒ x = 360° – 220° = 140°

c) The figure is having 5 sides. Hence, it is a pentagon.

Sum of angles of the pentagon = 540° Two angles at the bottom are linear pair.

∴, 180° – 70° = 110°

180° – 60° = 120°

⇒ 30° + 110° + 120° + x + x = 540°

⇒ 260° + 2x = 540°

⇒ 2x = 540° – 260° = 280°

⇒ 2x = 280°

= 140°

d) The figure is having 5 equal sides. Hence, it is a regular pentagon. Thus, its all angles are equal.

5x = 540°

⇒ x = 540°/5

⇒ x = 108°

**7.**

Solution:

a) Sum of all angles of triangle = 180°

One side of triangle = 180°- (90° + 30°) = 60°

x + 90° = 180° ⇒ x = 180° – 90° = 90°

y + 60° = 180° ⇒ y = 180° – 60° = 120°

z + 30° = 180° ⇒ z = 180° – 30° = 150°

x + y + z = 90° + 120° + 150° = 360°

b) Sum of all angles of quadrilateral = 360°

One side of quadrilateral = 360°- (60° + 80° + 120°) = 360° – 260° = 100°

x + 120° = 180° ⇒ x = 180° – 120° = 60°

y + 80° = 180° ⇒ y = 180° – 80° = 100°

z + 60° = 180° ⇒ z = 180° – 60° = 120°

w + 100° = 180° ⇒ w = 180° – 100° = 80°

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