This document provides a 3-sentence summary of a mathematics textbook chapter on polygons: The chapter discusses different types of polygons based on their number of sides, such as triangles having three sides and pentagons having five sides. It also presents formulas for calculating the sum of interior angles in polygons by dividing them into simpler shapes. Regular polygons are defined as those with all sides of equal length and all interior angles equal.

1. MATHEMATICS
DIGITAL TEXT BOOK
CLASSIX
submittedby,
AKHILA.V.V
B.Ed. Mathematics
18014376002Chapter 1
POLYGONS
A triangle hasthree sidesthree angles;aquadrilateral hasfour.
A Pentagonisa figure of five sidesandfive angles;six sides andsix anglesfromahexagon.

2. There are othernameslike this:a figure of sevensidesandanglesiscalledaheptagon;andof eight
sidesand,an octagon.There isa commonname forall these together:Polygons.
Sumof the angles
We knowthatthe sumof the anglesof a trianlesis1800 andthat the sum of the anglesof a
quadriateral is3600.
To compute the sum of the anglesof a pentagon.SplititintoPOLYGONS
Sumof the angles
Exteriorangles
Regularpolygons
2: PAIRSOFEQUATIONS
Algebraicdescription
Unificationquadriateral andtriangle.

3. Sumof the anglesof the pentagon
= sumof the anglesof the quadriateral
+ sumof the anglesof the triangle
= 3600+1800
= 5400
Generallyspeaking,we candivide anypolygon intopreviouspolygon(meaningone sideless)anda
triangle;andso itssumof angleswill be 1800 more thanthe sumof the anglesof the previousone.
Thus we have a sheme forcomputingthe sumof the anglesof polygon:
Triangle1800
Quadriateral 1800 +1800 = 2x1800
Pentagon 1800 +(2x1800) = 3x1800Hexagon 1800 +(3x1800) = 4x1800
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F The sumof the anglesof an sidedpolygonis(n-2) x 1800
Can youfindthe answersto these questions?
. The sum of the anglesof polygonis21600. What is the sum of the anglesof polygonwithone
more side?Whatabout a polygonwithone side less?
. What isthe sumof the anglesof polygonwith102 sides?
Exteriorangles
Look at thisfigure: Regularpolygons
Polygons,withequal anglesandlengthof side alsoequal are calledregularpolygons.
The figure belowshowaregular pentagonanda regularhexagon: Prove that the linesADandBD
divide ÐCDEintothe equal parts.
Since ÐCDE is angle of a regularpentagon,itisequal to1080,as seenearlies.

4. So,what isit that we wantto prove?
We needonlyshowthateachof ÐEDA, ÐADB andÐBDC is360.
How dowe findÐEDA?
It isan angle of DAED.What all thingsdo we know aboutthistriangle?
We haveÐAED= 1800
sinceitisangle of a regularpentagon.
So,
ÐEDA + ÐEAD = 1800 - 1080 = 720
Also,inthistriangle AE= ED (why?)
So,
ÐEDA = ÐEAD
We are earlierthatthe sum of these equal anglesi 720 So,each mustbe 300. Thus
ÐEDA = 360
Usingthe same argumentinDABCD,we get
ÐCDB = 360
(Try it)
Nowif we use the fact that
ÐEDA + ÐADB + ÐCDB = ÐEDC = 1080 andthat
ÐEDA = 300 = ÐCDB
We can easilyget
ÐADB = 360
Here are some problemsforyou:
. How muchis eachangle of a 12- sidesregularpolygon?
How muchis eachof itsextentangle?
. In the figure belowABCDEFisregularhexagon. Chapter2

5. PAIRSOF EQUATIONS
Let’sstart with a problemwhichyoucanddo in yourhead:
. The booksof the sameprice anda penfor five rupeeswere boughtforthirtyfive rupees.Whatis
the price of one book?
Look at thisproblem:
. Four chairsof the same price and a table for3150 rupeeswere boughtfor4950 rupees.Whatis
the price of a chair?
Here alsowe can mentallyworkoutthe steps;butforthe actual computationwe needpenand
paper,right?
Price of four chairs= 4950 - 3150 = Rs. 1800
Price of one chair = 1800 = Rs. 450
4
Algebraicdescription
Suppose we are toldthis:
The perimeterof a rectangle is20 centimetres.
Usingthis factalone,can we findthe lengthandbreath?
If we denote the lengthbyx andthe breadthby y, then
2 (x+y) = 20
and simplifythis,we get
x+y = 10
From this,we cannotfindthe lengthandbreadth.
Nowsuppose we knowone more factabout the lengthandbreadth:
Lengthe isI centimetre more thanbreadth.
Thisalsowe can write inalgebra,usingourearliernotation:
x = y+1

6. Nowcan we findlengthandbreadth?Forthat we must findthe numbersx andy such that boththe
equations
x + y = 10
x = y + 1
are true
By the secondequation,the numberx andy +1 are the same;andby the firstequation,the sumof the
numbersx andyis 10. So,the sum of the numbersy+1 and y is10.
That is
(y+ 1) + y = 10
From this,we get
2y + 1 = 10
and so
y = 10-1 = 4.5
2
Thus the breadthis4.5 centimetres.
What baoutthe length?
Since x + y = 10, we find
x = 10 - y = 10 - 4.5 = 5.5
so,the lengthis 5.5 centimetres
Nowtry these problems.
. Th numberof girlsin a classis 4 more thanthe numberof boys.Onaday whenonly8 boyswere
absent,the numberof girlswere double the numberof boys.How manygirlsand boysare there inthe
class?
. Th price of a table anda chair togetheris2500 rupees.The price of 3 tablesand4 chairsof this
kindis8000 rupees.Whatis the price of each.
Unification

7. Look at thisbroble:
The price od 2 pensand 3 note booksis 29 rupees;for2 pensand 5 note books,itis43. What is
the price of each?
Denotingthe price of a penby x and the price of a note bookby,ye,we can the givenpiecesof
informationtofromthe equations
2x + 3y = 29
2x + 5y = 43
Nowthese are to be combinedtoa single equationwithonlyone letter. So,subtracting29from 43 is
the same subtracting2x + 3y from2x+5y)
That is
(2x+ 5y) - (2x + 3y) = 43- 29,
Simplyfyingthis,we get
2y =14
fromwhichwe get
y = 7
Now, fromthe firstequation,we canfind
2x=29- 3y = 29 - (3x7) 8
and thisgives
x = 4
Nowtry these problemonyourown:
. Some autotrikshawsandbikesare parkedin a place.
These are 19 in all.The total numberof wheelsis45.
How manyof eachvehicle are there?
. Four timesanumberaddedto thrice anothernumbergives43.Twice the secondnumber,
substractedfromthrice the firstgives11. What are the numbers?