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**CBSE NCERT Solutions of Class 8
Mathematics **

### Chapter 3 Understanding Quadrilaterals

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__NCERT solutions
of Understanding Quadrilaterals Exercise 3.1__

__NCERT solutions of 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) Simple Curve: 1, 2, 5, 6, 7

(b) Simple Closed Curve: 1, 2, 5, 6, 7

(c) Polygon: 1, 2, 4

(d) Convex Polygon: 2

(e) Concave Polygon: 1, 4

Question 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 has 2
diagonals.

(b) A regular Hexagon has 9
diagonals.

(c) A triangle has no diagonal.

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!)

__Solution__:

__Case I__

The
sum of the measures of the angles of a convex quadrilateral is 360° because a
convex quadrilateral is made of two triangles. In the adjacent figure of convex
quadrilateral ABCD, it is made of two triangles ΔABD and ΔBCD. Therefore, the
sum of the measures of all the interior angles of this quadrilateral will be
same as the sum of all the interior angles of these two triangles = 360°

__Case II__

Yes,
this property also holds true even if the quadrilateral is not convex since any
quadrilateral can be divided into two triangles. In the adjacent figure the
concave quadrilateral ABCD, is made of two triangles ΔABD and ΔBCD. Hence, the sum
of all the interior angles of this concave quadrilateral is also 360°.

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__: A regular polygon is both ‘equiangular’ and ‘equilateral’. In other words, 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.

__Solution__:

(a)
Sum of all interior angles of a quadrilateral is 360º.

Therefore,

50°
+ 130° + 120° +

__/__= 360°*x*__/x__= 60°

(b)

From
the figure, it can be concluded that,

90º
+ y = 180º (Linear pair)

y =
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)

70
+ y = 180° (Linear pair)

y =
110°

60°
+ z = 180° (Linear pair)

z =
120°

Now,
consider pentagon,

120°
+ 110° + 30° +

*x + x*= 540°
(Sum
of all interior angles of a pentagon is 540º)

⇒ 260° + 2

*x*= 540°
⇒ 2

*x*= 280°
⇒

*x*= 140°
(d)

Given
figure is of a regular pentagon which is equilateral and equiangular.

Hence,
sum of all interior angles of this pentagon is 540º.

5

*x*= 540°*x*= 108°

Question 7:

(a) find

*x*+*y*+*z*(b) find*x*+*y*+*z*+*w*__Solution__:

(a) x + 90° = 180° (Linear pair)

x = 90°

Similarly, z = 150°

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

y = 120°

∴ x + y + z =
90° + 120° + 150° = 360°

(b) Sum of the measures of all
interior angles of a quadrilateral is 360

^{O}.
∴ a + 60° +
80° + 120° = 360°

⇒ a = 100°

And, x + 120° = 180° (Linear
pair)

⇒ x = 60°

Similarly, y = 100°, z = 120° and
w = 80°

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

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