Class 8 - Maths - 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

(a) Simple curve

(b) Simple closed curve

(c) Polygons

(d) Convex polygons

(e) Concave polygon

Question 2:

How many diagonals does each of the following have?

(a) A convex quadrilateral                  (b) A regular hexagon               (c) A triangle

(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)

Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral

in two triangles.

Ð A + ÐB + Ð C + Ð D = Ð 1 + Ð 6 + Ð 5 + Ð 4 + Ð 3 + Ð 2

= (Ð 1 + Ð 2 + Ð 3) + (Ð 4 + Ð 5 + Ð 6)

= 1800 + 1800         [By Angle sum property of triangle]

= 3600

Hence, the sum of measures of the triangles of a convex quadrilateral is 3600.

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 Δ ABD, Ð 1 + Ð 2 + Ð 3 = 1800 …………….1

In Δ BDC, Ð 4 + Ð 5 + Ð 6 = 1800 …………….2

Adding equation 1 and 2, we get

Ð 1 + Ð 2 + Ð 3 + Ð 4 + Ð 5 + Ð 6 = 1800 + 1800

=> Ð 1 + Ð 2 + Ð 3 + Ð 4 + Ð 5 + Ð 6 = 3600

=> Ð A + ÐB + Ð C + Ð D = 3600

Hence  proved.

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) When n = 7, then

Angle sum of a polygon = (n - 2) * 1800 = (7 - 2) * 1800 = 5 * 1800 = 9000

(b) When n = 8, then

Angle sum of a polygon = (n - 2) * 1800 = (8 - 2) * 1800 = 6 * 1800 = 10800

(c) When n = 10, then

Angle sum of a polygon = (n - 2) * 1800 = (10 - 2) * 1800 = 8 * 1800 = 14400

(d) When n = n, then

Angle sum of a polygon = (n - 2) * 1800

Question 5:

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

(a) 3 sides                         (b) 4 sides                            (c) 6 sides

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.

Question 6:

Find the angle measures x in the following figures:

(a) Using angle sum property of a quadrilateral,

500 + 1300 + 1200 + 3600 + x = 3600

=> 3000 + x = 3600

=> x = 3600 – 3000

=> x = 600

(b) Using angle sum property of a quadrilateral,

900 + 600 + 700 + x = 3600

=> 2200 + x = 3600

=> x = 3600 - 2200

=> x = 1400

(c) First base interior angle = 1800 – 700 = 1100

Second base interior angle = 1800 - 600 = 1200

There are 5 sides, n = 5

Angle sum of a polygon = (n - 2) * 1800

= (5 - 2) * 1800

= 3 * 1800

= 5400

So, 300 + x + 1100 + 1200 + x = 5400

=> 2600 + 2x = 5400

=> 2x = 5400 – 2600

=> 2x = 2800

=> x = 2800/2

=> x = 1400

(d) Angle sum of a polygon = (n - 2) * 1800

= (5 - 2) * 1800

= 3 * 1800

= 5400

Now, x + x + x + x + x = 5400

=> 5x = 5400

=> x = 5400/5

=> x = 1080

Hence each interior angle is 1080.

Question 7:

(a) Find x + y + z                              (b) Find x + y + z + w

(a) Since sum of linear pair angles is 1800.

So, 900 + x =1800

=> x= 1800 – 900 = 900

And z + 300 = 1800

=> z = 1800 – 300 = 1500

Also y = 900 + 300 = 1200             [Exterior angle property]

So, x + y + x = 900 + 1200 + 1500 = 3600

(b) Using angle sum property of a quadrilateral,

600 + 800 + 1200 + n = 3600

=> 2600 + n = 3600

=> n = 3600 - 2600

=> n = 1000

Since sum of linear pair angles is 1800.

So, w + 1000 = 1800 ……….(i)

x + 1200 = 1800 ……………..(ii)

y + 800 = 1800 ………………(iii)

z + 600 = 1800 ………………(iv)

Adding equation (i), (ii), (iii) and (iv), we get

x + y + z + w + 1000 + 1200 + 800 + 600 = 1800 + 1800 + 1800 + 1800

=> x + y + z + w + 3600 = 5400

=> x + y + z + w = 5400 – 3600

=> x + y + z + w = 3600

Exercise 3.2

Question 1:

Find x in the following figures:

(a) Here, 1250 + m =1800           [Linear pair]

=> m = 1800 – 1250 = 550

And 1250 + n = 1800                   [Linear pair]

=> n = 1800 – 1250 = 550

Since Exterior angle x = Sum of opposite interior angles

=> x = 550 + 550 = 1100

(b) Sum of angles of a pentagon = (n - 2) * 1800

= (5 - 2) * 1800

= 3 * 1800

= 5400

By linear pairs of angles,

Ð1 + 900 = 1800 ……….(i)

Ð2 + 600 = 1800 ……….(ii)

Ð3 + 900 = 1800 ……….(iii)

Ð4 + 700 = 1800 ……….(iv)

Ð5 + 1800 = x ……….(v)

Adding equation (i), (ii), (iii), (iv) and (v), we get.

x  + Ð1 + Ð2 + Ð3 + Ð4 + Ð5 + 3100 = 9000

=> x + 5400 + 3100 = 9000

=> x + 8500 = 9000

=> x = 9000 – 8500 = 850

Question 2:

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

(a) 9 sides                                     (b) 15 sides

(i) Sum of angles of a regular polygon = (n - 2) * 1800

= (9 - 2) * 1800

= 7 * 1800

= 12600

Each interior angle = Sum of interior angles/Number of Sides

= 12600/9 = 1400

Each exterior angle = 1800 - 1400 = 400

(ii) Sum of exterior angles of a regular polygon = 3600

Each interior angle = Sum of interior angles/Number of sides

= 3600/15 = 240

Question 3:

How many sides does a regular polygon have, if the measure of an exterior angle is 240?

Let number of sides is n.

Sum of exterior angles of a regular polygon = 3600

Number of sides = Sum of exterior angles/Each interior angle

= 3600/240 = 15

Hence, the regular polygon has 15 sides.

Question 4:

How many sides does a regular polygon have if each of its interior angles is 1650?

Let number of sides be n.

Exterior angle = 1800 - 1650 = 150

Sum of exterior angles of a regular polygon = 3600

Number of sides = Sum of exterior angles/Each interior angle

= 3600/150 = 24

Hence, the regular polygon has 24 sides.

Question 5:

(a) Is it possible to have a regular polygon with of each exterior angle as 220?

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

(a) No. (Since 22 is not a divisor of 3600)

(b) No, (Because each exterior angle is 1800 – 220 = 1580, which is not a divisor of 3600)

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?

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

interior angle of 600.

Sum of all the angles of a triangle = 1800

=> x + x + x = 1800

=> 3x = 1800

=> x = 1800/3

=> x = 600

(b) By (a), we can observe that the greatest exterior angle is 1800 - 600 = 1200.

Exercise 3.3

Question 1:

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

(i) AD = ____________________                (ii) Ð DCB = ____________________                          (iii) OC = ____________________               (iv) mÐ DAB + mÐ CDA = ____________________

(i) AD = BC                                              [Since opposite sides of a parallelogram are equal]

(ii) Ð DCB = Ð DAB                               [Since opposite angles of a parallelogram are equal]

(iii) OC = OA                                           [Since diagonals of a parallelogram bisect each other]

(iv) mÐ DAB + mÐ CDA = 1800          [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 30 = 300 in figure (iii)

(i) ÐB + ÐC = 1800                  [Adjacent angles in a parallelogram are supplementary]

=> 1000 + x = 180                                              0

=> x = 1800 – 1000 = 800

And z = x = 800                       [Since opposite angles of a parallelogram are equal]

Also, y = 1000                         [Since opposite angles of a parallelogram are equal]

(ii) x + 500 = 1800                   [Adjacent angles in a parallelogram are supplementary]

=> x = 1800 - 500 = 1300

And z = x = 1300                           [Corresponding angles]

(iii) x = 900                                     [Vertically opposite angles]

Now, y + 900 + 300 = 1800          [Angle sum property of a triangle]

=> y + 1200 = 1800

=> y = 1800 – 1200

=> y = 600

Also, z = y = 600                          [Alternate angles]

(iv) z = 800                                   [Corresponding angles]

Now, x + 800 = 1800                   [Adjacent angles in a parallelogram are supplementary]

=> x = 1800 - 800 = 1000

And y = 800             [Opposite angles are equal in a parallelogram]

(v) y = 1120                                 [Opposite angles are equal in a parallelogram]

=> 400 + y + x = 1800                 [Angle sum property of a triangle]

=> 400 + 1120 + x = 1800

=> 1520 + x = 1800

=> x = 1800 - 1520 = 280

And z = x = 280                         [Alternate angles]

Question 3:

Can a quadrilateral ABCD be a parallelogram, if:

(i) ÐD + ÐB = 1800?

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

(iii) Ð A = 70 and Ð C = 650?

(i)  ÐD + ÐB = 1800

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. Ð A ≠ Ð C

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.

ABCD is a quadrilateral in which angles

Ð A = Ð C = 1100

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.

Let two adjacent angles be 3x and 2x.

Since the adjacent angles in a parallelogram are supplementary.

So, 3x + 2x = 1800

=> 5x = 1800

=> x = 1800/5

=> x = 360

One angle = 3x = 3 * 360 = 1080

and another angle = 2x = 2 * 360 = 720

Question 6:

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

Let each adjacent angle be x.

Since the adjacent angles in a parallelogram are supplementary.

So, x + x = 1800

=> 2x = 1800

=> x = 1800/2

=> x = 900

Hence, each angle is 900

So, x + x + x = 1800

=> 3x = 1800

=> x = 1800/3

=> x = 600

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.

Here ÐHOP + 700 = 1800                           [Angles of linear pair]

=> Ð HOP = 1800 - 700 = 1100

And Ð E = Ð HOP                                     [Opposite angles of a ||gm are equal]

=> x = 1100

ÐPHE = Ð HPO                                         [Alternate angles]

So, y = 400

Now Ð EHO + Ð O = 700                                     [Corresponding angles]

=> 400 + z =700

=> z = 700 - 400 = 300

Hence, x = 1100, y = 400 and z = 300

Question 8:

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

(i) In parallelogram GUNS,

GS = UN                                       [Opposite sides of parallelogram are equal]

=> 3x = 18

=> x = 18/3 = 6 cm

Also GU = SN                              [Opposite sides of parallelogram are equal]

=> 3y – 1 = 26

=> 3y = 26 + 1

=> 3y = 27

=> y = 27/3

=> y = 9 cm

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

(ii) In parallelogram RUNS,

y + 7 = 20                                         [Diagonals of parallelogram bisects each other]

=> y = 20 - 7 = 13 cm

And x + y = 16

=> x + 13 = 16

=> x = 16 – 13

=> x = 3

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

Question 9:

In the figure, both RISK and CLUE are parallelograms. Find the value of x.

In parallelogram RISK,

Ð RIS = Ð K = 1200                       [Opposite angles of a parallelogram are equal]

Ðm + 1200 = 1800                        [Linear pair]

=> Ðm = 1800 - 1200 = 600

And Ð ECI = Ð L = 700               [Corresponding angles]

=> Ðm + Ðn + ÐECI = 1800            [Angle sum property of a triangle]

=> 600 + Ðn + 700 = 1800

=> 1300 + Ðn = 1800

=> Ðn = 1800 – 1300 = 500

Also x = Ðn = 500                              [Vertically opposite angles]

Question 10:

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

Here, Ð M + Ð L = 1000 + 800 = 1800                [Sum of interior opposite angles is 1800]

So, NM and KL are parallel.

Hence, KLMN is a trapezium.

Question 11:

1. Find mÐ C in figure , if 𝐴𝐵 || 𝐷𝐶

Here, Ð B + Ð C = 1800 [Since 𝐴𝐵 || 𝐷𝐶]

=> 120 + mÐC = 1800

=> m ÐC = 1800 – 1200 = 600

Question 12:

Find the measure of Ð P and Ð S if 𝑆𝑃 || 𝑅𝑄 in given figure.

(If you find mÐ R is there more than one method to find mÐ P)

Here,

Ð P + Ð Q = 1800                           [Sum of co-interior angles is 1800]

=> Ð P + 1300 = 1800

=> Ð P = 1800 – 1300

=> Ð P = 500

Since Ð R = 900                             [Given]

So, ÐS + 900 = 1800

=> ÐS = 1800 – 900

=> ÐS = 900

Yes, there is one more method is there to find ÐP.

ÐS + Ð R + Ð Q + Ð P = 3600                        [Angle sum property of quadrilateral]

=> 900 + 900 + 1300 + ÐP = 3600

=> 3100 + ÐP = 3600

=> ÐP = 3600 - 3100

=> ÐP = 500

Exercise 3.4

Question 1:

State whether true or false:

(a) All rectangles are squares.

(b) All rhombuses are parallelograms.

(c) All squares are rhombuses and also rectangles.

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

(a) False. Since, squares have all sides are equal.

(b) True. Since, in rhombus, opposite angles are equal and diagonals intersect at mid-point.

(c) True. Since, squares have the same property of rhombus but not a rectangle.

(d) False. Since, all squares have the same property of parallelogram.

(e) False. Since, all kites do not have equal sides.

(f) True. Since, all rhombuses have equal sides and diagonals bisect each other.

(g) True. Since, trapezium has only two parallel sides.

(h) True. Since, all squares have also two parallel lines.

Question 2:

Identify all the quadrilaterals that have:

(a) four sides of equal lengths.                                (b) four right angles.

(a) Rhombus and square have sides of equal length.

(b) Square and rectangle have four right angles.

Question 3:

Explain how a square is:

(i) a quadrilateral            (ii) a parallelogram            (iii) a rhombus         (iv) a rectangle

(i) A square is a quadrilateral, if it has four unequal lengths of sides.

(ii) A square is a parallelogram, since it contains both pairs of opposite sides equal.

(iii) A square is already a rhombus. Since, it has four equal sides and diagonals bisect at 900 to

each other.

(iv) A square is a parallelogram, since having each adjacent angle a right angle and opposite

sides are equal.

Question 4:

(i) bisect each other.

(ii) are perpendicular bisectors of each other.

(iii) are equal.

(i) If diagonals of a quadrilateral bisect each other then it is a rhombus, parallelogram,

rectangle or square.

(ii) If diagonals of a quadrilateral are perpendicular bisector of each other, then it is a rhombus

or square.

(iii) If diagonals are equal, then it is a square or rectangle.

Question 5:

Explain why a rectangle is a convex quadrilateral.

A rectangle is a convex quadrilateral since its vertex are raised and both of its diagonals lie in

its interior.

Question 6:

ABC is a right-angled triangle and O is the mid-point of the side opposite to the right angle. Explain why O is

Since, two right triangles make a rectangle where O is equidistant point from A, B, C and D

because O is the mid-point of the two diagonals of a rectangle.

Since AC and BD are equal diagonals and intersect at mid-point.

So, O is the equidistant from A, B, C and D.