# 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

Sol :

(a) 1, 2, 5, 6, 7

(b) 1, 2, 5, 6, 7

(c) 1, 2

(d) 2

(e) 1

Question 2

How many diagonals does each of the following have?

(a) A convex quadrilateral

Sol :

There are 2 diagonals in a convex quadrilateral.

(b) A regular hexagon

Sol :

There are 9 diagonals in a regular hexagon.

(c) A triangle

Sol :

A triangle does not have any diagonal in it.

Question 3

What is the sum of the measures of the angels of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

Sol :

The sum of the measures of the angles of a convex quadrilateral is 360° as a convex quadrilateral is made of two triangles.

Here, ABCD is a convex quadrilateral, made of two triangles ΔABD and ΔBCD. Therefore, the sum of all the interior angles of this quadrilateral will be same as the sum of all the interior angles of these two triangles i.e., 180º + 180º = 360º

Yes, this property also holds true for a quadrilateral which is not convex. This is because any quadrilateral can be divided into two triangles.

Here again, ABCD is a concave quadrilateral, made of two triangles ΔABD and ΔBCD. Therefore, sum of all the interior angles of this quadrilateral will also be 180º + 180º = 360º

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 × 180° = (4 − 2) × 180° 3 × 180° = (5 − 2) × 180° 4 × 180° = (6 − 2) × 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

Sol :

From the table, it can be observed that the angle sum of a convex polygon of n sides is (n −2) × 180º. Hence, the angle sum of the convex polygons having number of sides as above will be as follows.

(a) (7 − 2) × 180º = 900°

(b) (8 − 2) × 180º = 1080°

(c) (10 − 2) × 180º = 1440°

(d) (n − 2) × 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

Sol :

A polygon with equal sides and equal angles is called a regular polygon.

(i) Equilateral Triangle

(ii) Square

(iii) Regular Hexagon

Question 6

Find the angle measure x in the following figures.

 (a) (b) (c) (d)

Sol :

(a)

Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,

50° + 130° + 120° + x = 360°

300° + x = 360°

x = 60°

(b)

From the figure, it can be concluded that,

90º + a = 180º (Linear pair)

a = 180º − 90º = 90º

Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,

60° + 70° + x + 90° = 360°

220° + x = 360°

x = 140°

(c)

From the figure, it can be concluded that,

70 + a = 180° (Linear pair)

a = 110°

60° + b = 180° (Linear pair)

b = 120°

Sum of the measures of all interior angles of a pentagon is 540º.

Therefore, in the given pentagon,

120° + 110° + 30° + x + x = 540°

260° + 2x = 540°

2x = 280°

x = 140°

(d)

Sum of the measures of all interior angles of a pentagon is 540º.

5x = 540°

x = 108°

Question 7

(a) find x + y + z

Sol :

(a) x + 90° = 180° (Linear pair)

x = 90°

z + 30° = 180° (Linear pair)

z = 150°

y = 90° + 30° (Exterior angle theorem)

y = 120°

x + y + z = 90° + 120° + 150° = 360°

(b) find x + y + z + w

Sol :

Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,

a + 60° + 80° + 120° = 360°

a + 260° = 360°

a = 100°

x + 120° = 180° (Linear pair)

x = 60°

y + 80° = 180° (Linear pair)

y = 100°

z + 60° = 180° (Linear pair)

z = 120°

w + 100° = 180° (Linear pair)

w = 80°

Sum of the measures of all interior angles = x + y + z + w

= 60° + 100° + 120° + 80°

= 360°