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Understanding Quadrilaterals
Parallelogram: A quadrilateralwith each pair ofopposite sidesparallel.
Properties:
(1) Opposite sides are equal.(2) Opposite angles are equal.
(3) Diagonals bisect one another.
Rhombus: A parallelogram with sidesof equal length.
Properties:
(1) All the properties of a parallelogram.(2) Diagonals are perpendicular to each other.
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Rectangle: A parallelogramwith a right angle.
Properties:
(1) All the properties of a parallelogram.(2) Each of the angles is a right angle.
(3) Diagonals are equal.
Square: A rectangle with sides of equal length.
Properties: All properties of rectangle.
Kite: A quadrilateral with exactly two pairs of equal consecutive sides
Properties:(1) The diagonals are perpendicular
to one another(2) One of the diagonals bisects the other.
(3) In the figure DB but CA
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Diagonal: A diagonal is a line connecting two no-consecutive vertices of a polygon.
Angle Sum of a Polygon: The angle sum of a polygon is given by following formula:
(n-2)180, where n is the number of sides of a polygon.
Sum of exterior angles of a polygon: This is always 360, no matter what is the
number of sides in a given polygon. Take example of a triangle. All vertex of a tri-angle will make to exterior angles. As you know the angle sum of a triangle is 180,
so three exterior angles will sum up to 180 and two sets of three exterior angleseach will sum up to 360. In a rectangle all exterior angles are of 90, hence their
sum is equal to 360.
EXERCISE 1
1. How many diagonals does each of the following have?
(a) A convex quadrilateral (b) A regular hexagon (c) A triangle
Answer:(a) 2 (b) 9 (c) 0
2. What is the sum of the measures of the angles of a convex quadrilateral? Will this
property hold if the quadrilateral is not convex?
Answer: Using the formula:(n-2)180= (4-2)180 = 360.
For concave quadrilateral also angle sum will be same because number of sides issame.
3. What can you say about the angle sum of a convex polygon with number of sides?(a) 7 (b) 8 (c) 10
Answer:
(a) Number of sides = 7
So, Angle Sum= 900180)27(180)2(n(b) Number of sides = 8
So, Angle Sum= 1080180)28((c ) Number of sides = 10
So, Angle Sum = 1440180)210(4. What is a regular polygon? State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
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Answer: In a regular polygon all sides are of equal length.
(i) Triangle (ii) Quadrilateral (iii) Hexagon
5. Find the angle measurexin the following figures.
(a)
Answer: As we know that angle sum of a quadrilateral is 360.So, 50+130+120+x= 360
Or, x= 360-300 = 60
(b)
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Answer: Angles on one side of a line always add up to 180, so third angle in the
given quadrilateral is 90
Now, x+70+60+90 = 360Or, x= 360-220 = 140
(c)
Answer: Angle adjacent to 70 = 180-70 = 110Angle adjacent to 60 = 180-60 =120
Now, 2x+110+120+30 = 360Or, 2x= 360-260 = 100
Or, x= 50
(d)
Answer: Angle sum of pentagon(n-2)180
= (5-2)180=3 X 180 = 540
Hence, measurement of 1 angle of a pentagon= 540 5 = 108(e) ?zyx
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Answer: As we know sum of all external angle of a polygon is always 360.
So, 360zyxAlternate Method:The angle adjacent to z = 180-(90+30) = 60
Now, x= 180-90 = 90
y= 180-30 = 150 z= 180 - 60 = 120
Now, 36012015090zyx
(f) ?zyxw
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Answer: Sum of all the external angles of a polygon is always 360.
Hence, 360zyxwEXERCISE 2
1. Findxin the following figures.(a)
Answer: 110)125125(360x Because sum of all exterior angles of a polygon is always 360(b)
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Answer: Number of Sides =11
317
22
360360 ==
gleexteriorAn
As the final answer is not a whole number so there is no possibility of a polygon with
1 exterior angle measuring 22.
(b) Can it be an interior angle of a regular polygon? Why?
Answer: If interior angle is 22 then the exterior angle = 180-22=158
On dividing 360 by 158 we cant get answer in whole number, so such a polygon isnot possible.
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?
Answer: The polygon with minimum number of sides is a triangle, and each angle ofan equilateral triangle measures 60, so 60 is the minimum value of the possible in-terior angle for a regular polygon. For an equilateral triangle the exterior angle is
180-60=120 and this is the maximum possible value of an exterior angle for a
regular polygon.
EXERCISE 31. Given a parallelogram ABCD. Complete each statement along with the definition or
property used.
(i) AD = Opposite Sides are Equal
(ii) DCB = Opposite Angles are equal(iii) OC = Diagonals Bisect Each Other
(iv) m DAB + m CDA = 180
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2. Consider the following parallelograms. Find the values of the unknowns x, y, z.
(i)
Answer: 80100180xAs Opposite angles are equal in a parallelogram
So, 100yAnd, 80z(ii)
Answer:x, y and zwill be complementary to 50.
So, Required angle=180-50=130
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(iii)
Answer:z being opposite angle= 80x and yare complementary, x and y=180-80=100
(iv)
Answer: As angles on one side of a line are always complementarySo, x=90 y= 180-(90+30)=60The top vertex angle of the above figure = 60 2=120Hence, bottom vertex Angle = 120 andz=60
(v)
Answer:y= 112, as opposite angles are equal in a parallelogram
x= 180-(112-40)=28
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As adjacent angles are complementary so angle of the bottom left vertex=180-112=68
So, z=68-40=28
Another way of solving this is as follows:As angles x and zare alternate angles of a transversal so they are equal in measure-
ment.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?Answer: (i)It can be , but not always as you need to look for other criteria as well.
(ii) In a parallelogram opposite sides are always equal, here AD BC, so its not aparallelogram.
(iii) Here opposite angles are not equal, so it is not a parallelogram.
5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Findthe measure of each of the angles of the parallelogram.
Answer: Opposite angles of a parallelogram are always add upto 180.
So, xx 23180 +1805x
36xSo angles are; 108336 and 72236
6. Two adjacent angles of a parallelogram have equal measure. Find the measure of
each of the angles of the parallelogram.
Answer: 90, as they add up to 180
7. The adjacent figure HOPE is a parallelogram. Find the angle measuresx, yandz. State the properties you use to find them.
Answer: Angle opposite to y= 180-70=110
Hence, y= 110x=180-(110+40)=30, (triangles angle sum)
z=30 (Alternate angle of a transversal)
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8. The following figures GUNS and RUNS are parallelograms. Findxand y. (Lengthsare in cm)
Answer: As opposite sides are equal in a parallelogramSO, 2613 =y
273 =y9y
Similarly, 183 =x6x
Answer: As you know diagonals bisect each other in a parallelogram.
So, 207 =y13720 =y
Now, 16yx
1613 =x31316 =x
9. In the given figure both RISK and CLUE are parallelograms. Find the value ofx.
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Answer: In parallelogram RISK
60120180ISK
Similarly, in parallelogram CLUE
11070180CEUNow, in the triangle
10)60110(180xEXERCISE 41. State whether True or False.
(a) All rectangles are squaresAll squares are rectangles but all rectangles cant be squares, so this state-
ment is false.
(b) All kites are rhombuses.
All rhombuses are kites but all kites cant be rhombus.
(c) All rhombuses are parallelograms
True.
(d) All rhombuses are kites.
True
(e) All squares are rhombuses and also rectangles
True; squares fulfill all criteria of being a rectangle because all angles are
right angle and opposite sides are equal. Similarly, they fulfill all criteria ofa rhombus, as all sides are equal and their diagonals bisect each other.
(f) All parallelograms are trapeziums.
False; All trapeziums are parallelograms, but all parallelograms cant betrapezoid.
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(g) All squares are not parallelograms.
False; all squares are parallelograms
(h) All squares are trapeziums.
True;
2. Identify all the quadrilaterals that have.(a) four sides of equal length (b) four right angles
Answer: (a) If all four sides are equal then it can be either a square or a rhombus.
(b) All four right angles make it either a rectangle or a square.
3. Explain how a square is.
(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle
Answer: (i) Having four sides makes it a quadrilateral
(ii) Opposite sides are parallel so it is a parallelogram
(iii) Diagonals bisect each other so it is a rhombus(iv) Opposite sides are equal and angles are right angles so it is a rectangle.
4. Name the quadrilaterals whose diagonals.(i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal
Answer: Rhombus; because, in a square or rectangle diagonals dont intersect at
right angles.
5. Explain why a rectangle is a convex quadrilateral.
Answer: Both diagonals lie in its interior, so it is a convex quadrilateral.
6. ABC is a right-angled triangle and O is the mid point of the side opposite to theright angle. Explain why O is equidistant from A, B and C.
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Answer: If we extend BO to D, we get a rectangle ABCD. Now AC and BD are diag-
onals of the rectangle. In a rectangle diagonals are equal and bisect each other.
So, AC= BDAO= OC
BO= ODAnd AO=OC=BO=OD
So, it is clear that O is equidistant from A, B and C.
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