# Class 8 NCERT Solutions – Chapter 3 Understanding Quadrilaterals – Exercise 3.1

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?

#### Solution:

Here we can see that only two diagonals are possible for the convex quadrilateral.

### (b) A regular hexagon

#### Solution:

Here we can see that only three diagonals are possible for the convex quadrilateral.

### (c) A triangle

#### Solution:

In case of a triangle no diagonal is possible.

### 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?

#### Solution:

Sum of the angles of a quadrilateral is always 180Â°. It doesn’t depend on whether the quadrilateral is a convex or a concave.

### What can you say about the angle sum of a convex polygon with number of sides?

#### Solution:

From the above given table we can deduce that for a given number of sides n the sum of the angles of a convex polygon is (n – 2) Ã— 180Â°.

a) 7

Ans: (7 – 2) Ã— 180Â° = 5 Ã— 180Â° = 900Â°

b) 8

Ans: (8 – 2) Ã— 180Â° = 6 Ã— 180Â° = 1080Â°

c) 10

Ans: (10 – 2) Ã— 180Â° = 8 Ã— 180Â° = 1440Â°

d) n

Ans: (n – 2) x 180Â°

### 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:

Regular Polygon is a polygon with equal sides and equal angles.

(i) 3 sides : Equilateral triangle

(ii) 4 sides : Square

(iii) 6 sides : Regular Hexagon

### Question 6. Find the angle measure x in the following figures.

#### Solution:

a) Total sum of the angles of the quadrilateral = 360Â°

Therefore, 50Â° + 130Â° + 120Â° + x = 360Â° â‡’ 300Â° + x = 360Â° â‡’ x = 60Â°

b) Total sum of the angles of the quadrilateral = 360Â°

Therefore, 60Â° + 70Â° + 90Â° + x = 360Â° â‡’ 220Â° + x = 360Â° â‡’ x = 140Â°

c) Total sum of the angles of the polygon = 540Â°

Angles adjacent to 60Â° = 180Â° – 60Â° = 120Â°

Angles adjacent to 70Â° = 180Â° – 70Â° = 110Â°

Therefore, 110Â° + 30Â° + 120Â° + x + x= 540Â° â‡’ 260Â° + 2x = 540Â° â‡’ 2x = 280Â° â‡’ x = 140Â°

d) Total sum of the angles of the polygon = 540Â°

Therefore, xÂ° + xÂ° + xÂ° + xÂ° + xÂ°= 540Â° â‡’ 5x = 540Â° â‡’ x = 108Â°

### Question 7.  (a) Find x + y + z                                  (b) Find x + y + z + w

#### Solution:

(a) By exterior angle property, y = 90Â° + 30Â° = 120Â°

x = 180Â° – 90Â° = 90Â°

z = 180Â° – 30Â° = 150Â°

So, the answer x + y + z = 360Â°

(b) Angle adjacent to w = 360Â° – ( 60Â° + 80Â° + 120Â° ) = 100Â°

w = 180Â° – 100Â°  = 80Â°

x = 180Â° – 120Â°  = 60Â°

y = 180Â° – 80Â° = 100Â°

z = 180Â° – 60Â° = 120Â°

So, the answer w + x + y + z = 360Â°

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