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FATHIMA MEMORIAL

TRAINING COLLEGE,

DIGITAL TEXT BOOK

NAME OF STUDENT TEACHER: ANEESA.A

SUBJECT :

MATHEMATICS

CANDIDATE CODE :

18014352001

1


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ATHE ATICS

STANDARD : IX

2


3/21

CONTENT

PAGE

1. Polygons

3

2. Sum of angles

4

3. Exterior Angles7

4. n!"anging sum #

$. %i&erent 'in(s of

3


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Polygons

11

OL!GONS

Polygons

)e "a*e learn a+out triangles an( ,ua(rilaterals. A triangle "as t"ree si(es

an( t"ree angles- a ,ua(rilaterals "as four.

Polygonsare manysi(e( /gures0 it" si(es t"at are line segments. Polygons

are name( a!!or(ing to t"e num+er of si(es an( angles t"ey "a*e. T"e most

familiar olygons are t"e triangle0 t"e re!tangle0 an( t"e s,uare .Polygons

also "a*e (iagonals0 "i!" are segments t"at oin to *erti!es an( are not

si(es.

)e !an !lassify olygons a!!or(ing to t"e num+er of si(es or *erti!es.

T"e simle olygon is a triangle. A triangle "as t"ree si(es t"us0 it is a t"ree

si(e( olygon.

4
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A four si(e( olygon is !alle( a ,ua(rilateral. A /*e si(e( olygon is !alle( a

entagon.

n t"is manner0 e !an o+tain a sixsi(e( olygon !alle( a "exagon0 a

se*en si(e( olygon 0!alle( a "etagon0 an( so on.

Ea!" si(e of a olygon is !onne!te( +y to !onse!uti*e *erti!es of t"e

olygon.

A (iagonal is a line segment t"at !onne!ts t"e non !onse!uti*e *erti!es of a

olygon.

Polygons are 2-dimensional shapes. They are made of straight lines,

and the shape is "closed

Polygon

(straight sides)

Nota Polygon

(has a curve)

Nota Polygon

(open, not closed)

S"# $% &'( )*+(-

$


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)e 'no t"at t"e sum of t"e angles of a triangle is 156an( t"at t"e sum of

t"e angles of a ,ua(rilateral is 386.

Start it" *ertex Aan( !onne!t it to all ot"er *erti!es 9it is alrea(y !onne!te( to :

E +y t"e si(es of t"e /gure;. T"ree triangles are forme(. T"e sum of t"e angles in

triangle !ontains 156156 ? $46

Noti!e t"at a entagon "as 5si(es0 an( t"at 3triangles ere forme( +y !onne!ting

*erti!es. T"e num+er of triangles forme( ill +e 2less t"an t"e num+er of si(es.

Geometrical

figure

No.o

f

sides

Figure showing

triangles

No. of

triangles

Sum of the

angles

Triangle

! !"#$

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%uadrilatera

l

& ''!"#*#$

Pentagon

+ !"#+$

e-agon

* & &!"#'#$

@eresenting t"e num+er of si(es of a olygon as n0 t"e num+er of triangles forme

Sin!e ea!" triangle

!ontains 1800 t"e sum

t"e interior angles of a

olygon is 180n/ 2.

Example 1 =)"at is t"e sum of t"e angles of a olygon it" 162 si(es

7

Sum of Interior Angles

of a Polygon= 180(n - 2)(where n = number of sides)


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Bere t"e num+er of si(es of a olygon is 162.9.e- n?162;

Sum of t"e angles of a olygon ?9n2;156

? 91622; > 156

? 166 > 156

? 1566.

!am"le 2# ow many sides does a polygon have if the sumof its interiorangles is '#/0

Since, the num1er of degrees is given, set the formula a1ove e2ual to '#/,

and solve for n$

1

80(n - 2) '#

n3 ' &

n = 6

Exercise

:

T"e sum of t"e angles of a olygon is 2186. )"at is t"e sum of t"e angles of

olygon it" one more si(e )"at a+out a olygon it" one si(e less

T"e sum of t"e angles of a olygon is 2766. Bo many si(es (oes it "a*e

%oes any olygon "a*e t"e sum of its angles e,ual to 1666 Bo a+out #66

A 16 si(e( olygon "as all its angles e,ual. Bo mu!" is ea!" angle

Ea!" angle of a olygon is 1$6. Bo many si(es (oes it "a*e

E&($ )*+(-

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Doo' at t"is /gure=

!

A : P

n A:C 0 t"e si(e A: is exten(e( an( t"is ro(u!es a ne angle outsi(e t"e triang

T"is angle C:P is !alle( an external angle of t"e triangle.

)"at is t"e relation +eteen t"is angle an( t"e 9internal; angle C:A of t"e triangle

C

FNN

A : P

)"en e a(( u t"e interior Angle an( Exterior Angle e get a straig"t line0 156.

HC:P ? 156 H C:A

No instea( of A: 0 if e exten( C:0 t"en also e get an exterior angle at :.

C

#


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A : P

I

)"at is t"e relation +eteen t"e external Angle A:I an( HC:P

T"ey are t"e oosite angles ma(e +y t"e lines AP an( CI interse!ting at :.

So HA:I ?HC:P

As it" :0 e !an (ra external angles at ea!" of t"e ot"er to *erti!es also.

An( li'e t"is 0 e !an (ra external angles at ea!" *ertex of any olygo

Jor examle 0 loo' at t"e external angles of a ,ua(rilateral=

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No try to (o t"ese ro+lems=

T"e angles of a triangle are 360 46 an( 116. Jin( t"e measures of its

external

Angles.

T"ree angles of a ,ua(rilateral are 860 7$ an( 166. Jin( t"e fourt"

angle.

Also /n( all t"e four external angles.

U*')*+*+ -"#

)e "a*e learnt a tri!' to !omute t"e sum of t"e angles of a olygon. s

t"ere a ay to !omute t"e sum of t"e external angles

DetKs start it" a triangle= C

A :

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12/21

T"e sum of t"e external angles at A an( t"e angle of t"e triangle itself at A0is

156.T"e same t"ing "aens at t"e *erti!es : an( C also.

So0 t"e sum of t"ese t"ree airs of external an( internal angles is

3180=540 . T"at is0 t"e sum of t"e t"ree external angles an( t"e t"ree

angles of t"e triangle is $46

. n t"is0 t"e sum of t"e angles of t"e triangle is156. Jrom t"is0 e see t"at t"e sum of t"e external angles only is

540 180 =360 .

)"at a+out a ,ua(rilateral0 instea( of a triangle

Bere0 at ea!" of t"e four *erti!es0 t"ere is a linear air forme( +y an

external angle an( an angle of t"e ,ua(rilateral. T"e sum of t"e angles in

ea!" su!" air is 156

. So0 t"e four external angles an( t"e four angles of

t"e ,ua(rilateral toget"er ma'e 4180 =720 .

n t"is0 t"e sum of t"e four angles of t"e ,ua(rilateral is 386. So0 t"e

sum of t"e four external angles is 720360 =360 .

DetKs t"in' a+out an n si(e( olygon in general. T"ere are n *erti!es in

all- an( at ea!" *ertex0 a linear air forme( +y an external angle an( an

angle of t"e olygon itself. So0 sum of all t"ese angles is n180 . n t"is0

t"e sum of t"e angles of t"e olygon is (n2 )180 .

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So0 sum of t"e external angles n180 [(n2 )180 ]

2180

360

Jin(ing t"e sum of t"e exterior angles of a olygon is simle. No matter "at

tye of a olygon you "a*e0 t"e sum of t"e exterior angles is alays e,ual to

386.

The sum of the external angles of any polygon is 3!.

No try t"ese ro+lems0

Pro*e t"at in any triangle0 ea!" external angle is e,ual to t"e sum of t"e

nternal angles at t"e ot"er to *erti!es.

n 12 si(e( olygon0 t"e external angles are all e,ual. Bo mu!" is

ea!" external angle An( ea!" angle of t"e olygon

T"e sum of t"e angles of a olygon an( t"e sum of its external angles

are e,ual. Bo many si(es (oes it "a*e

Ea!" external angle of a olygon is 26. Bo many si(es (oes it "a*e

D6((*& K*7- $% $+$*-

#on$ex and #onca$e Polygons

E*ery olygon is eit"er convexor concave.T"e (i&eren!e +eteen !on*ex

an( !on!a*e olygons lies in t"e measures of t"eir angles.

Jor a olygon to +e !on*ex0 all of its interior angles must +e less t"an 156

(egrees. Ot"erise0 t"e olygon is !on!a*e. Anot"er ay to t"in' of it is t"is=

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t"e (iagonals of a !on*ex olygon ill +e in t"e interior of t"e olygon0

"ereas !ertain (iagonals of a !on!a*e olygon ill lie outsi(e t"e olygon0

on its exterior. :elo in Part A are some !on*ex olygons0 an( in Part :0

some !on!a*e olygons.

Simple and complex polygon

A -#9(olygon "as only one +oun(ary0 an( it (oesnLt !ross o*er itself.

A $#9(olygon interse!ts itselfM any rules a+out olygons (onLt or'

"en it is !omlex.

Simple Polygon

(this one4s a Pentagon)

5omple- Polygon

(also a Pentagon)

%egular Polygons

f all angles of a triangle are e,ual0 "o mu!" is ea!" angle

Sin!e t"e angles are e,ual0 t"e si(es of t"is triangle must also +e e,ual.

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15/21

On t"e ot"er "an(0 "at if t"e si(es of a triangle are all e,ual T"en its

angles are also e,ual. 9 Su!" triangles are !alle( e,uilateral triangles0 rig"t;

No if t"e angles of a ,ua(rilateral are all e,ual0 is it ne!essary t"at its si(es

are also e,ual

n any re!tangle0 t"e angles are all e,ual - +ut t"e si(es may not +e e,ual. f

t"e si(es are also e,ual 0 it +e!omes a s,uare.

Polygons li'e t"is0 it" e,ual angles an( lengt"s of si(es also e,ual are

!alle( (+") 9$+$*-.

f all angles are e,ual an( all si(es are e,ual0 t"en it is (+")0 ot"erise it

is(+")

Polygons !an also +e !lassi/e( as e,uilateral0 e,uiangular0 or +ot".

E,uilateral olygons "a*e !ongruent si(es0 li'e a r"om+us. E,uiangular

olygons "a*e !ongruent interior angles0 li'e a re!tangle. )"en a olygon is

+ot" e,uilateral an( e,uiangular0 it is !alle( a regular polygon. A s,uare is

an examle of a regular olygon. T"e !enter of a regular olygon is t"e oint

1$

6egular 7rregular


16/21

from "i!" all t"e *erti!es of t"e olygon are e,ui(istant. @egular olygons

"a*e se!ial roerties t"at eLll exlore in t"e next se!tion. :elo are some

examles of e,uiangular0 e,uilateral0 an( regular olygons.

$9(&(- $% ) (+") 9$+$*-

&pothem'in radius(

T"e aot"em of a olygon is a line from t"e !entre to t"e mi(oint of a si(e.

T"is is also t"e inra(iust"e ra(ius of t"e in!ir!le of t"e olygon. Jor a

olygon of n si(es0 t"ere are n ossi+le aot"ems0 all t"e same lengt" of

!ourse.

%adius 'circumradius(

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17/21

T"e ra(ius of a regular olygon is a line from t"e !enter to any *ertex. t ill+e t"e same for any *ertex. T"e ra(ius is also t"e ra(ius of t"e olygonKs

!ir!um!ir!le0 "i!" is t"e !ir!le t"at asses t"roug" e*ery *ertex. n t"is

role0 it is sometimes !alle( t"e !ir!umra(ius.

rregular olygons are not usually t"oug"t of as "a*ing a !enter or ras(ius.

)ncircle

T"e in!ir!le is t"e largest !ir!le t"at ill /t insi(e a olygon t"at tou!"es

e*ery si(e.

f t"e num+er of si(es is 30 t"is is an e,uilateral triangle an( its in!ir!le is

exa!tly t"e same as t"e one (es!ri+e( in in!ir!le of a Triangle.

T"e inra(ius of a regular olygon is exa!tly t"e same as its aot"em.

#ircumcircle

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18/21

T"e !ir!le t"at asses t"roug" ea!" *ertex of t"e regular olygon. f t"e

num+er of si(es is 30 t"en t"e result is an e,uilateral triangle an( its

!ir!um!ir!le is exa!tly t"e same as t"e one (es!ri+e( in Cir!um!ir!le of a

Triangle.

& polygon is regular *hen all angles are e+ual and all

sides are e+ual.

T"e /gures +elo s"o a regular entagon an( a regular "exagon=

Bo mu!" is ea!" angle of a regular entagon

T"e sum of t"e angles is 3180 =540 ; an( sin!e it is regular0 t"is is t"e

sum of /*e e,ual angles.

So0 ea!" angle is1

5540 =108 .

Similarly 0 e !an easily see t"at ea!" angle of a regular "exagon is

1

64180 =120

Doo' at anot"er examle=

%

15


19/21

E C

A :

ABCDEis a regular entagon.

Pro*e t"at t"e linesADan( BD (i*i(e


20/21

%ra a "exagon it" all angles e,ual0 +ut not all si(es e,ual.

Bo mu!" is ea!" angle of a 12 si(e( regular olygon Bo mu!" is

ea!" of its external angle

Pro*e t"at in a regular entagon 0 t"e eren(i!ular from any *ertex to

t"e oosite si(e +ise!ts t"at si(e.

n t"e /gure +elo0ABCDE" is a regular "exagon.

E %

J C

A :

Pro*e t"at ABCDE" is a re!tangle.

n t"e /gure +elo0 A:C%EJ is a regular "exagon.

E %

J C

A :

Pro*e t"at ACE is an e,uilateral triangle.

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Bo mu!" is an angle of a 38 si(e( regular olygon

One angle of a regular olygon is 144. Bo many si(es (oes it "a*e

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