**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

(b) Simple closed curve**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 two diagonals.

Here, AC and BD are two diagonals.

(b) A regular hexagon has 9 diagonals.

Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.

(c) A triangle has no diagonal.**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 :**

Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.

∠A + ∠B + ∠C + ∠D

= ∠1 + ∠6 + ∠5 + ∠4 + ∠3 + ∠2

= (∠1 + ∠2 + ∠3) + (∠4 + ∠5 + ∠6)

[By Angle sum property of triangle]

= **360º**

Hence, the sum of measures of the triangles of a convex quadrilateral is **360º**

Yes, if quadrilateral is not convex then, this property will also be applied.

Let ABCD is a non-convex quadrilateral and join BD, which also divides the quadrilateral in two triangles.

Using angle sum property of triangle,

In △ABD, ∠1 + ∠2 + ∠3 = 180º;……….(i)

In △BDC, ∠4 + ∠5 + ∠6 = 180º;……….(i)

Adding eq. (i) and (ii),

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360º

⇒;∠1 + ∠2 + (∠3 + ∠4) + ∠5 + ∠6

= 360º

⇒;∠A + ∠B + ∠C + ∠D = 360º

Hence proved.**Question 4.**

Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)**Solution :**

(a) When n = 7, then

Angle sum of a polygon =

(b) When n= 8, then

Angle sum of a polygon =

(c) When n = 10, then

Angle sum of a polygon =

(d) When n =** n** then

Angle sum of a polygon = **Question 5.**

What is a regular polygon? State the name of a regular polygon of:

(a) 3 sides

(b) 4 sides

(c) 6 sides**Solution :**

A regular polygon: A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon.

(i) 3 sides

Polygon having three sides is called a triangle.

(ii) 4 sides

Polygon having four sides is called a quadrilateral.

(iii) 6 sides

Polygon having six sides is called a hexagon.

x = 140º

(d)

**NCERT Solutions for Class 8 Maths Exercise 3.2**

**Question 1.**

Find** x**;in the following figures:**Solution :**

(a) Here,

[Linear pair]

By linear pairs of angles,

Adding eq. (i), (ii), (iii), (iv) and (v),

**Question 2.**

Find the measure of each exterior angle of a regular polygon of:

(a) 9 sides

(b) 15 sides**Solution :**

(i) Sum of angles of a regular polygon =

=

Each interior angle =

Each exterior angle =

(ii) Sum of exterior angles of a regular polygon = **360º**

Each interior angle =

**Question 3.**

How many sides does a regular polygon have, if the measure of an exterior angle is 24º?

**Solution :**

Let number of sides be **n.**

Sum of exterior angles of a regular polygon = **360º**

Number of sides =

Hence, the regular polygon has 15 sides.

**Question 4.**

How many sides does a regular polygon have if each of its interior angles is 165º?**Solution :**

Let number of sides be** n.**

Exterior angle =

Sum of exterior angles of a regular polygon = **360º**

Number of sides =

Hence, the regular polygon has 24 sides.**Question 5.**

(a) Is it possible to have a regular polygon with of each exterior angle as** 22º?**

(b) Can it be an interior angle of a regular polygon? Why?**Solution :**

(a) No. (Since 22 is not a divisor of **360º**)

(b) No, (Because each exterior angle is **180º – 22º = 158º**; which is not a divisor of **360º**)

**Question 6.**

(a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?**Solution :**

(a) The equilateral triangle being a regular polygon of 3 sides has the least measure of an

interior angle of **60º,**

∵ Sum of all the angles of a triangle

= **180º**

∴; x + x +x = 180**º**

⇒ 3 x = 180**º**

⇒ x = 60**º**

(b) By (a), we can observe that the greatest exterior angle is **180º – 60º**

= 120**º**.

**NCERT Solutions for Class 8 Maths Exercise 3.3**

**Question 1.**

Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(i) AD = _______________

(ii) ∠DCB = ______________

(iii) OC = _____________

(iv) m∠DAB + m∠CDA = ________

**Solution :**

(i) AD = BC

[Since opposite sides of a parallelogram are equal]

(ii) ∠DCB = ∠DAB

[Since opposite angles of a parallelogram are equal]

(iii) OC = OA

[Since diagonals of a parallelogram bisect each other]

(iv) m∠DAB + m∠CDA = **180º**

[Adjacent angles in a parallelogram are supplementary]

**Question 2.**

Consider the following parallelograms. Find the values of the unknowns **x , y ,z**

Note: For getting correct answer, read **3º = 30º** in figure (iii)**Solution :**

(i) ∠B + ∠C = 180**º**

[Adjacent angles in a parallelogram are supplementary]

[Since opposite angles of a parallelogram are equal]

Also y = 100º

[Since opposite angles of a parallelogram are equal]

(ii) x + 50º = 180º x = 50º = 180º

[Adjacent angles in a ||gm are supplementary]

[Corresponding angles]

(iii) x = 90º

[Vertically opposite angles]

[Adjacent angles in a ||gm are supplementary]

[Opposite angles are equal in a ||gm]

(v) y = 112º

[Opposite angles are equal in a gm]

[Alternate angles]

**Question 3.**

Can a quadrilateral ABCD be a parallelogram, if:

(i) ∠D + ∠B = 180º

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii) ∠A = 70º and ∠C = 65º?

**Solution :**

(i) ∠D + ∠B = 180º

It can be, but here, it needs not to be.

(ii) No, in this case because one pair of opposite sides are equal and another pair of opposite sides are unequal. So, it is not a parallelogram.

(iii) No.

Since opposite angles are equal in parallelogram and here opposite angles are not equal in quadrilateral ABCD. Therefore it is not a parallelogram.

**Question 4.**

Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measures.

**Solution :**

ABCD is a quadrilateral in which angles A = C = 110º

Therefore, it could be a kite.

**Question 5.**

The measure of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

**Solution :**

Let two adjacent angles be 3X and 2X

Since the adjacent angles in a parallelogram are supplementary.

**Question 6.**

Two adjacent angles of a parallelogram have equal measure. Find the measure of the angles of the parallelogram.

**Solution :** Let each adjacent angle be **x,**

Since the adjacent angles in a parallelogram are supplementary.

Hence, each adjacent angle is 90º

**Question 7.**

The adjacent figure HOPW is a parallelogram. Find the angle measures x , y and z State the properties you use to find them.

∠HOP + 70º = 180º

**Question 8.**

The following figures GUNS and RUNS are parallelograms. Find** x** and **y **(Lengths are in cm)**Solution :**

(i) In parallelogram GUNS,

GS = UN

[Opposite sides of parallelogram are equal]

Also GU = SN

[Opposite sides of parallelogram are equal]

Hence, x= 6 cm and x= 9 cm.

(ii) In parallelogram RUNS,

⇒ x = 16 -13

⇒ x = 16 -13

⇒ x = 3cm

Hence, x = 3 cm and y = 13cm.

**Question 9.**

In the figure, both RISK and CLUE are parallelograms. Find the value of **Solution :**

In parallelogram RISK,

[Vertically opposite angles]

**Question 10.**

Explain how this figure is a trapezium. Which is its two sides are parallel?**Solution :**

Here, ∠M + ∠L = 100º + 80º = 180º

[Sum of interior opposite angles is 180º]

NM and KL are parallel.

Hence, KLMN is a trapezium.**Solution :**

**Question 12.**

Find the measure of ∠P and ∠S if in given figure.

(If you find m∠R is there more than one method to find m∠P)**Solution :**

**NCERT Solutions for Class 8 Maths Exercise 3.4**