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

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**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 polygonbecause polygon is a simple closed curve made up of

line segments and the 4^{th}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°

► ∠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,** 1****80 ^{∘}+180^{∘} = 360**

**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 **

**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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