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# NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

## Class 8 : NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

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

Q.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
Solution.

• Simple curve: A simple curve is a curve that does not cross itself.
• Simple closed curve: In simple closed curves the shapes are closed by line-segments or by curved lines.
• Polygon: A simple closed curve made up of only line segments is called a polygon.
• Convex polygon: A Convex polygon is defined as a polygon with no portions of their diagonals in their exteriors.
• Concave Polygon: A concave polygon is defined as a polygon with one or more interior angles greater than 180°.

(a) Simple curves are: (1), (2), (5), (6) and (7).
(b) Simple closed curves are: (1), (2), (5), (6) and (7).
(c) Polygons are: (1), (2)

Note

• '4' is not a polygon because polygon is a simple closed curve made up of
line segments and the 4th figure is not a simple curve because it crosses itself.
• Therefore, '4' is not a polygon.
• In the latest edition, this figure is no longer listed as the polygon.

(d) Convex polygon is: (2)
(e) Convex polygon is: (1)

Q.2. How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
Solution.
A diagonal is a line segment connecting two non-consecutive vertices of a polygon. Draw the above-given polygon and mark vertices and then, draw lines joining the two non-consecutive vertices. From this, we can calculate the number of diagonals.
(a) Convex quadrilateral
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 = Q.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 ΔABC and ΔADC. We know that the sum of the angles of a triangle is 180 degree. So,
∠6+∠5+∠4 = 180° (Sum of the angles of ΔABC is 180°)
∠1+∠2+∠3 = 180° (Sum of the angles of ΔADC is 180°)
Adding the above equations, we get:
∠6+∠5+∠4+∠1+∠2+∠3 = 360°
On Rearranging the terms:
∠6+∠1+∠3+∠4+∠5+∠2 = 360°
∠A+∠C+∠B+∠D = 360° (∠6+∠1 = ∠A, ∠3+∠4 = ∠C  )
Hence, the sum of measures of the triangles of a convex quadrilateral is 360.

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

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?
(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 – 2) x 180°
(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 – 2) x 180° = (7 – 2) x 180°
= 5 x 180° = 900°

(b) When n = 8
Substituting n = 8 in the above formula, we have:
Sum of interior angles of a polygon having 8 sides
= (n – 2) x 180° = (8 – 2) x 180°
= 6 x 180° = 1080°

(c) When n = 10
Substituting n = 10 in the above formula, we have:

Sum of interior angles of a polygon having 10 sides
= (n – 2) x 180°
= (10 – 2) x 180°
= 8 x 180° = 1440°

(d) When n = n
The sum of interior angles of a polygon having n-sides = (n – 2) x 180°

Q.5. What is a regular polygon?
State the name of a regular polygon of following sides
(a) 3 sides
(b) 4 sides
(c) 6 sides

Solution.
A polygon is said to be a regular polygon if:

• The measures of its interior angles are equal and
• The lengths of its sides are equal

(a) A regular polygon of three sides is called an equilateral Triangle. (b) A regular polygon of 4 sides is called square. (c) A regular polygon of 6 sides is called regular hexagon. Q.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 = 360°
x + 120° + 130° + 50° = 360°
⇒ x + 300° = 360°
x = 360° – 300° = 60°

(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 = 360°
The figure is having four sides. Hence, it is a quadrilateral.

x + 60° + 70° + 90° = 360°
x + 220° = 360°
x = 360° – 220° = 140°

(c) The figure is having 5 sides. Hence, it is a pentagon.
Sum of interior angles of a pentagon = 540°
Two angles at the bottom are forming linear pair.
∴ 180° - 70° = 110°
180° – 60° = 120°
Interior angles are: 30°, x°, x°, 110° and 120°
30° + x + x + 110° + 120° = 540°
2x + 260° = 540°
2x = 280°
x = 140° (d) The figure is having 5 equal sides. Hence, It is a regular pentagon. Thus, it's all angles are equal.
Sum of all interior angles of a regular pentagon = 540°.

x + x + x + x + x = 540°
5x = 540°
x = 540° ÷ 5 = 108°

Q.7.  (a) Find x + y + z.
(b) Find x + y + z + w.
Solution.
(a) Sum of all the angles of a triangle = 180°

∴ One angle of a triangle is 180° - (90°+30°) = 60°
⇒ x + 90° = 180° (Linear pair)
⇒ x = 180° – 90° = 90°
⇒ y = 30° + 90° = 120° (∵ Sum of interior opposite angles = exterior angle)

⇒ z = 180° – 30° = 150° (Linear pair)
Now, x + y + z = 90° + 120° + 150° = 360°

(b) Sum of interior angles of a quadrilateral = 360°
∴ ∠1 + 120° + 80° + 60° = 360°
∠1 + 260° = 360°
1 = 360° – 260° = 100° Now, x + 120° = 180° (Linear pair)
∴ x = 180° – 120° = 60°
y + 80° = 180°   (Linear pair)
∴  y = 180° – 80° = 100°
z + 60 = 180° (Linear pair)
∴  z = 180° – 60° = 120°
w + 100 = 180°  (Linear pair)
∴ w = 180° – 100° = 80°
Thus, x + y + z + w = 60° + 100° + 120° + 80° = 360°

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## Mathematics (Maths) Class 8

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