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1
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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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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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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