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INDIAN CENTRAL SCHOOL,KUWAIT

1. CLASS : VIII

2. SUBJECT:MATHEMATICS

3. TOPIC : UNDERSTANDING

QUADRILATERALS

4. CHAPTER : 3

5. SLOT: 8

6. CLASS WORK

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UNDERSTANDING QUADRILATERLS

Polygon:

A simple closed curve made up of only line segments is called a polygon. Each

straight line in a polygon is called its side.

A polygon is called triangle, quadrilateral, Pentagon, Hexagon, Heptagon,

Octagon, Nonagon and Decagon according as it contains 3,4,5,6,7,8,9and10 sides

respectively.

There are two types of Polygons:

1) Concave polygons:

A polygon in which at least one angle is more than 180˚ is called a concave

polygon.

2) Convex Polygon:

A polygon in which each angle is less than 180˚is called a convex polygon.

Regular polygon:

A polygon having all sides equal and all angles equal is called a regular polygon.

Eg: Square

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Irregular Polygons:

Polygons which are not regular are called irregular Polygons.

Eg: Rectangle

Quadrilateral:

A quadrilateral is a closed figure bounded by four line segments. It has 4 sides, 4

vertices and 4 angles.

Angle sum property of a quadrilateral:

Sum of all the angles of a quadrilateral is equal to 360˚

Note:

For a regular polygon of n sides , we have

i) each exterior angle =(360

𝑛)˚

ii) each interior angle = 180˚- (each exterior angle)

iii) Number of diagonals in a polygon of n sides =𝑛(𝑛−3)

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Ex 3.1

1.Given here are some figures: (Homework)

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

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

Number of diagonals =𝑛(𝑛−3)

2 =

4(4−3)

2 =2

(b) A regular hexagon

Number of diagonals =𝑛(𝑛−3)

2 ==

6(6−3)

2 = 9

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(c) A triangle

Number of diagonals =𝑛(𝑛−3)

2=

3(3−3)

2 = 0

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: (a) 7 Angle sum = (7−2)×180⁰ = 5×180⁰=900⁰ (b) 8 Angle sum = (8−2)×180⁰ = 6×180⁰=1080⁰

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(c) 10 Angle sum = (10−2)×180⁰ = 8×180⁰=1440⁰ (d) n Angle sum = (n−2)×180⁰ Note: Sum of all interior angles of a polygon of n sides = (n-2)x 180° 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 polygon, which is both ‘equilateral’ and ‘equiangular’, is called a regular polygon. (i) 3 sides The name of the regular polygon of 3 sides is an equilateral triangle. (ii) 4 sides The name of the regular polygon of 4 sides is square (iii) 6 sides The name of the regular polygon of 6 sides is a regular hexagon

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6.Find the angle measure x in the following figures

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Solution:

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

(a) x = 180° - 90°= 90° (linear pair)

z= 180°- 30° = 150° (linear pair)

y= 30° +90° =120° (exterior angle property of a triangle)

Hence ,x+ y + z =90 °+150° +120° = 360°

(b) x=180 ⁰– 120⁰ = 60⁰ (linear pair)

Y= 180⁰ – 80 ⁰=100⁰ (linear pair)

Z= 180 ⁰– 60⁰ =120°(linear pair)

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<1 = 360 ⁰– (120 ⁰+80⁰ +60⁰) = 100⁰(Angle sum property )

w = 180 ⁰– 100 ⁰= 80⁰(linear pair )

Thus x + y+ z +w = 60⁰ +100 ⁰+120⁰ +80⁰ =360⁰

Note : The sum of the measures of the external angles of any polygon is 360⁰.

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EX 3.2

1.Find x in the following figures.

2.Find the measure of each exterior angle of a regular polygon of

(i) 9 sides

(ii) 15 sides

Solution:

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(i) 9 sides

Measure of each exterior angle=360°

9=40°

(ii) 15 sides

Measure of each exterior angle=360°

15=24°

3.How many sides does a regular polygon have if the measure of an exterior angle

is 24°?

Solution:

Let the number of sides be n.

⇒ n = 360°

24° =15

Hence, the number of sides is 15.

4.How many sides does a regular polygon have if each of its interior angles is

165°?

Solution:

∵ Each interior angle=165°

∴ Each exterior angle = 180°-165°=15° (linear pair property)

Let the number of sides be n ,then

n = 360°

15°= 24

Hence, the number of sides is 24.

5.(a) Is it possible to have a regular polygon with a measure of each exterior angle as 22°? (b) Can it be an interior angle of a regular polygon? Why? Solution: (a) No ; (since 22 is not a factor of 360). (b) No ; (because each exterior angle is 180° – 22° = 158°, which is not a factor of 360).

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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 is a regular polygon of 3 sides has the minimum measure of an interior angle = 60°. (b) By (a), we can see that the maximum exterior angle possible for a regular polygon is 180° – 60° = 120°.

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TYPES OF QUADRILATERALS:

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Trapezium:

A quadrilateral having exactly one pair of parallel sides is called a trapezium.

Here AD // BC.

Note :The adjacent angles in a parallelogram are supplementary.

EX 3.3

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 Opposite sides of a parallelogram are equal

(ii) ∠DCB = ∠DAB Opposite angles of a parallelogram are equal

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(iii) OC = OA ∵ Diagonals of a parallelogram bisect each other

iv) m∠DAB + m∠CDA = 180° ∵ Adjacent angles in a parallelogram are supplementary.

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

Solution:

(i) y = 100° (Opposite angles of a parallelogram are equal)

x + 100° = 180° (Adjacent angles in a parallelogram are supplementary)

⇒ x = 180° – 100°

⇒ x = 80°

⇒ z =x = 80° (Opposite angles of a parallelogram are of equal measure)

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(ii) x + 50° = 180° (Adjacent angles in a parallelogram are supplementary)

⇒ x = 180° – 50° = 130°

⇒ y = x = 130° (The opposite angles of a parallelogram are of equal measure)

⇒ x= z = 130° (corresponding angles )

(iii) x = 90° (Vertically opposite angles are equal)

x + y + 30° = 180° (By angle sum property of a triangle)

⇒ 90° + y + 30° = 180°

⇒ 120° + y = 180°

⇒ y = 180° – 120° = 60°

z =y = 60⁰ (alternate angles )

(iv) y = 80° (Opposite angles of a parallelogram are of equal measure)

x + 80° = 180° (Adjacent angles in a parallelogram are supplementary)

⇒ x = 180° – 80°

⇒ x = 100°

z = 80° (corresponding angles )

(v) y = 112° (Opposite angles of a parallelogram are equal)

x + y + 40° = 180° (By angle sum property of a triangle)

⇒ x + 112° + 40° = 180°

⇒ x + 152° = 180°

⇒ x = 180°- 152°

⇒ x = 28°

z = x = 28°.(Alternate interior angles)

3.Can a quadrilateral ABCD be a parallelogram if

(i) ∠D + ∠B = 180° ?

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

(iii) ∠A = 70° and ∠C = 65°?

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Solution:

(i) Can be, but need not be

(ii) No ; in a parallelogram, opposite sides are equal; but here, AD ≠ BC.

(iii) No ; in a parallelogram, opposite angles are of equal measure;

but here ∠A ≠ ∠C.

4.Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly

two opposite angles of equal measure.

Solution:

A kite, for example

5.The measures 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 the two adjacent angles be 3x° and 2x°.

Then,

3x° + 2x° = 180°( Adjacent angles in a parallelogram are supplementary)

⇒ 5x° = 180°

⇒ x⁰= 180°

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⇒ x° = 36°

⇒ 3x° = 3 x 36° = 108° and

2x° = 2 x 36° = 72°.

Since, the opposite angles of a parallelogram are of equal measure, therefore the

measures of the angles of the parallelogram are 72°, 108°, 72°, and 108°.

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

of each of the angles of the parallelogram.

Solution:

Let the two adjacent angles of a parallelogram be x° each.

Then,

x° + x° = 180°( Adjacent angles in a parallelogram are supplementary)

⇒ 2x° = 180°

⇒ x∘=180°

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⇒ x° = 90°.

Since the opposite angles of a parallelogram are of equal measure, therefore the

measure of each of the angles of the parallelogram is 90°, i.e., each angle of the

parallelogram is a right angle.

7.The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z.

State the properties you use to find them.

Solution:

x = 180° – 70° = 110°

(Linear pair property and the opposite angles of a parallelogram are of equal

measure.)

∵ HOPE is a || gm

∴ HE || OP and HP is a transversal

∴ y = 40° (alternate interior angles)

⇒ 40° + z + 110° = 180°(Angle sum property )

⇒ z + 150° = 180°

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⇒ z = 180° – 150°

⇒ z = 30°.

Hence x=110⁰ , y=40⁰ , z= 30⁰

8.The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths

are in cm)

(i)

(ii)

Solution:

(i) In figure GUNS

3x = 18 ( the opposite sides of a parallelogram are of equal length)

⇒ x = 18

3 = 6

Also ,

3y – 1 = 26 ( the opposite sides of a parallelogram are of equal length)

⇒ 3y = 26 + 1

⇒ 3y = 27

y= 27

3 = 9

Hence, x = 6cm; y = 9cm

(ii)In figure RUNS

Since the diagonals of a parallelogram bisect each other, therefore,

⇒ x + y = 16 …(1) and y + 7 = 20 …(2)

From (2),

⇒ y = 20 – 7 = 13

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Putting y = 13 in (1), we get

⇒ x + 13 = 16 ⇒ x = 16 – 13 = 3.

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

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

Solution:

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

Solution:

∵ ∠KLM + ∠NML = 80° + 100° = 180°

∴ KL || NM ( since the sum of consecutive interior angles is 180°)

∴ KLMN is a trapezium.

Its two sides KL and NM are parallel.

11.Find m∠C in the figure, if AB|| DC.

Solution:

∵ AB || DC

∴ m∠C + m∠B = 180° ( The sum of consecutive interior angles is 180°)

m∠C+ 120° = 180°

⇒ m∠C = 180° – 120° = 60°.

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12.Find the measure of ∠P and ∠S, if SP || RQ in the figure. (If you find m< R, is

there more than one method to find m∠P ?)

Solution:

∵ SP|| RQ

∴ m∠P+m∠Q = 180°

∵ The sum of consecutive interior angles is 180°

⇒ m∠P + 130° = 180°

⇒ m∠P = 180° – 130°

⇒ m∠P = 50°

Again, m∠R + m∠S = 180°

∵ The sum of consecutive interior angles is 180°

⇒ 90° + m < S = 180°

⇒ m∠S = 180° – 90° = 90°

Yes; there is one more method of finding m∠P if m∠R is given and that is by using

the angle sum property of a quadrilateral.

We have,

m∠P + m∠Q + m∠R + m∠S = 360°

⇒ m∠P + 130° + 90° + 90° = 360°

⇒ m∠P + 310° = 360°

⇒ m∠P = 360° – 310° = 50°.

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