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

Question 1. Given here are some figures:

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) 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     

Solution:

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

Answer: 2

(b) A regular hexagon

Solution:

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

Answer: 3

(c) A triangle

Solution:

In case of a triangle no diagonal is possible.

Answer: 0

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 180°. It doesn’t depend on whether the quadrilateral is a convex or a concave.

Question 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?

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) × 180^°.

a) 7

Ans: (7 – 2) × 180° = 5 × 180° = 900°

b) 8

Ans: (8 – 2) × 180° = 6 × 180° = 1080°

c) 10

Ans: (10 – 2) × 180° = 8 × 180° = 1440°

d) n

Ans: (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:

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 = 360^°

    Therefore, 50° + 130° + 120° + x = 360° ⇒ 300° + x = 360° ⇒ x = 60°

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

   Therefore, 60° + 70° + 90° + x = 360° ⇒ 220° + x = 360° ⇒ x = 140°

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

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

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

   Therefore, 110° + 30° + 120° + x + x= 540° ⇒ 260° + 2x = 540° ⇒ 2x = 280° ⇒ x = 140° 

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

   Therefore, x° + x° + x° + x° + x°= 540° ⇒ 5x = 540° ⇒ x = 108°

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

Solution: 

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

      x = 180° – 90° = 90°

      z = 180° – 30° = 150°

So, the answer x + y + z = 360°

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

By Adjacent Angle property,

     w = 180° – 100°  = 80°

     x = 180° – 120°  = 60°

     y = 180° – 80° = 100°

     z = 180° – 60° = 120°

So, the answer w + x + y + z = 360°

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