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

All you need of Class 8 at this link: Class 8

**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 **

Simple closed curve:

Polygon:

line segments and the 4

In the latest edition, this figure is no longer listed as the polygon.

(a) A convex quadrilateral

(b) A regular hexagon

A diagonal is a line segment connecting two

A convex quadrilateral has two diagonals

Here, AC and BD are two diagonals.**(b) Regular hexagon**

Here, the diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD. Totally there are **9** diagonals.**(c) A triangle**

A triangle has no diagonal because there no two non-consecutive vertices.**Note:** Number of diagonals in a polygon of n-sides =**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: **

ABCD is a convex quadrilateral made of two triangles Î”ABC and Î”ADC. We know that the sum of the angles of a triangle is 180 degree. So,

âˆ 6+âˆ 5+âˆ 4 = 180Â° (Sum of the angles of Î”ABC is 180Â°)

âˆ 1+âˆ 2+âˆ 3 = 180Â° (Sum of the angles of Î”ADC is 180Â°)

Adding the above equations, we get

âˆ 6+âˆ 5+âˆ 4+âˆ 1+âˆ 2+âˆ 3 = 360Â°

On Rearranging the terms,

âˆ 6+âˆ 1+âˆ 3+âˆ 4+âˆ 5+âˆ 2 = 360Â°

âˆ A+âˆ C+âˆ B+âˆ D = 360Â° (âˆ 6+âˆ 1 = âˆ A, âˆ 3+âˆ 4 = âˆ C )

Hence, the sum of measures of the triangles of a convex quadrilateral is 360^{âˆ˜}.

**Yes**, even if quadrilateral is not convex then, this property applies. Let ABCD be a non-convex quadrilateral; join BD, which also divides the quadrilateral in two triangles.

Using the angle sum property of triangle again, ABCD is a concave quadrilateral, made of two triangles Î”ABD and Î”BCD. Therefore, the sum of all the interior angles of this quadrilateral will also be,

180^{âˆ˜}+180^{âˆ˜} = 360^{âˆ˜}**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 | 180Â° | 2 x 180Â° = (4 â€“ 2) x 180Â° | 3 x 180Â° = (5 â€“ 2) x 180Â° | 4 x 180Â° = (6 â€“ 2) x 180Â° |

**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 â€“ 2) x 180Â°****(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 â€“ 2) x 180Â° = (7 â€“ 2) x 180Â°

= 5 x 180Â° = 900Â°

**(b)** When n = 8

Substituting n = 8 in the above formula, we have

Sum of interior angles of a polygon having 8 sides

= (n â€“ 2) x 180Â° = (8 â€“ 2) x 180Â°

= 6 x 180Â° = 1080Â°

**(c)** When n = 10

Substituting n = 10 in the above formula, we have

Sum of interior angles of a polygon having 10 sides

= (n â€“ 2) x 180Â°

= (10 â€“ 2) x 180Â°

= 8 x 180Â° = 1440Â°

(d) When n = n

The sum of interior angles of a polygon having n-sides = (n â€“ 2) x 180Â°**Question 5.**** What is a regular polygon?****. ****State the name of a regular polygon of following sides:(i) 3 sides (ii) 4 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**(i)** A regular polygon of three sides is called an **equilateral Triangle**.

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

**(iii)** A regular polygon of 6 sides is called **regular hexagon**.**Question 6. ****Find the angle measure x in the following figures.****Solution: ****(a) **The figure is having four sides. Hence, it is a quadrilateral.

As, Sum of interior angles of a quadrilateral = 360Â°

â‡’ x + 120Â° + 130Â° + 50Â° = 360Â°

â‡’ x + 300Â° = 360Â°

â‡’ x = 360Â° â€“ 300Â° = 60Â°

**(b) **The figure is having four sides. Hence, it is a quadrilateral. Also, one side is perpendicular.

As, sum of interior angles of a quadrilateral = 360Â°

The figure is having four sides. Hence, it is a quadrilateral.

â‡’ x + 60Â° + 70Â° + 90Â° = 360Â°

â‡’ x + 220Â° = 360Â°

â‡’ x = 360Â° â€“ 220Â° = 140Â°

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

Sum of interior angles of a pentagon = 540Â°

Two angles at the bottom are forming linear pair.

âˆ´ 180Â° - 70Â° = 110Â°

180Â° â€“ 60Â° = 120Â°

âˆ´ Interior angles are: 30Â°, xÂ°, xÂ°, 110Â° and 120Â°

â‡’ 30Â° + x + x + 110Â° + 120Â° = 540Â°

â‡’ 2x + 260Â° = 540Â°

â‡’ 2x = 280Â°

â‡’ x = 140Â°**(d)** The figure is having 5 equal sides. Hence, It is a regular pentagon. Thus, its al angles are equal.

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

â‡’ x + x + x + x + x = 540Â°

â‡’ 5x = 540Â°

â‡’ x = 540Â° Ã· 5 = 108Â°**Question 7.****(a) Find x + y + z. ****(b) Find x + y + z + w.****Solution:****(a)** Sum of all the angles of a triangle = 180Â°

âˆ´ One angle of a triangle is 180Â° - (90Â°+30Â°) = 60Â°

x + 90Â° = 180Â° (Linear pair)

â‡’ x = 180Â° â€“ 90Â° = 90Â°

y = 30Â° + 90Â° = 120Â° (âˆµ Sum of interior opposite angles = exterior angle)

z = 180Â° â€“ 30Â° = 150Â° (Linear pair)

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

**(b)** Sum of interior angles of a quadrilateral = 360Â°

âˆ´ âˆ 1 + 120Â° + 80Â° + 60Â° = 360Â°

â‡’ âˆ 1 + 260Â° = 360Â°

â‡’ âˆ 1 = 360Â° â€“ 260Â° = 100Â°

Now, x + 120Â° = 180Â° (Linear pair)

âˆ´ x = 180Â° â€“ 120Â° = 60Â°

y + 80Â° = 180Â° (Linear pair)

âˆ´ y = 180Â° â€“ 80Â° = 100Â°

z + 60 = 180Â° (Linear pair)

âˆ´ z = 180Â° â€“ 60Â° = 120Â°

w + 100 = 180Â° (Linear pair)

âˆ´ w = 180Â° â€“ 100Â° = 80Â°

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

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

212 videos|147 docs|48 tests

### NCERT Textbook- Understanding Quadrilaterals

- Doc | 20 pages
### Points to Remember- Understanding Quadrilaterals

- Doc | 5 pages
### Difference Between Regular and Irregular Polygons

- Video | 04:10 min
### Test: Understanding Quadrilaterals- 2

- Test | 20 ques | 20 min
### Understanding Convex and Concave Polygons

- Video | 03:52 min

- Test: Understanding Quadrilaterals- 1
- Test | 10 ques | 10 min
- Classification of Polygons based on Number of sides
- Video | 03:43 min