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Title: Class 10 Triangles
Description: These notes are very helpful for class 10 students
Description: These notes are very helpful for class 10 students
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ANSOLED
CLASS10
CHAPTER–6
TRI
ANGLES
WhatI
saTr
i
angl
e?
At
r
i
angl
ei
sapol
y
gonchar
act
er
i
zedbyt
hr
eeangl
esandt
hr
eesi
des
...
Tr
i
angl
esar
ecl
assi
f
i
edi
nt
odi
f
f
er
entt
y
pesbasedont
he
measur
esoft
hei
rangl
esandt
hel
engt
hsoft
hei
rsi
des
...
Scal
eneTr
i
angl
e:
At
r
i
angl
ewi
t
ht
hr
eesi
desofdi
f
f
er
entl
engt
hs
...
I
soscel
esTr
i
angl
e:
At
r
i
angl
ewi
t
hatl
eastt
wosi
desofequal
l
engt
h
...
Equi
l
at
er
al
Tr
i
angl
e:
At
r
i
angl
ewi
t
hal
l
t
hr
eesi
desofequal
l
engt
handal
l
t
hr
ee
angl
esmeasur
i
ng60degr
ees
...
Acut
eTr
i
angl
e:
At
r
i
angl
ewi
t
hal
l
angl
esmeasur
i
ngl
esst
han90degr
ees
...
Ri
ghtTr
i
angl
e:
At
r
i
angl
ewi
t
honeangl
emeasur
i
ngexact
l
y90degr
ees
...
Obt
useTr
i
angl
e:
At
r
i
angl
ewi
t
honeangl
emeasur
i
ngmor
et
han90degr
ees
...
Thei
rcor
r
espondi
ngangl
esar
eequal
...
Thei
rcor
r
espondi
ngsi
desar
ei
nt
hesamer
at
i
oorpr
opor
t
i
on
...
Si
deSi
deSi
de(
SSS)Si
mi
l
ar
i
t
yCr
i
t
er
i
on:
Whent
hecor
r
espondi
ngsi
desoft
wo
t
r
i
angl
esar
ei
nt
hesamer
at
i
o,
t
hei
rcor
r
espondi
ngangl
esar
eal
soequal
,
i
ndi
cat
i
ngsi
mi
l
ar
i
t
y
...
Angl
eAngl
eAngl
e(
AAA)Si
mi
l
ar
i
t
yCr
i
t
er
i
on:
I
ft
hecor
r
espondi
ngangl
esoft
wo
t
r
i
angl
esar
eequal
,
t
hei
rcor
r
espondi
ngsi
deswi
l
l
bei
nt
hesamer
at
i
o,
demonst
r
at
i
ngsi
mi
l
ar
i
t
y
...
Angl
eAngl
e(
AA)Si
mi
l
ar
i
t
yCr
i
t
er
i
on:
Whent
woangl
esofonet
r
i
angl
ear
eequal
t
ot
woangl
esofanot
hert
r
i
angl
e,
t
het
r
i
angl
esar
econsi
der
edsi
mi
l
ar
...
Si
deAngl
eSi
de(
SAS)Si
mi
l
ar
i
t
yCr
i
t
er
i
on:
I
foneangl
eofat
r
i
angl
ei
sequal
t
o
oneangl
eofanot
hert
r
i
angl
e,
andt
hesi
desi
ncl
udi
ngt
heseangl
esar
ei
nt
he
samer
at
i
o,
t
het
r
i
angl
esar
edeemedsi
mi
l
ar
...
”
Now,
l
etusunder
st
andt
heBasi
cpr
opor
t
i
onal
i
t
yt
heor
em wi
t
ht
hehel
pofadi
agr
am
...
Wedr
awal
i
nePQpar
al
l
elt
ot
he
si
deBCofABCandi
nt
er
sectt
hesi
desABandACi
nPandQ,
r
espect
i
v
el
y
...
Fori
nst
ance,
consi
dert
wot
r
i
angl
esΔABCandΔPQR:
Ther
ef
or
e,
basedont
hear
eaofsi
mi
l
art
r
i
angl
est
heor
em,
wecanexpr
ess:
Ar
eaoft
r
i
angl
eABC/
Ar
eaofTr
i
angl
ePQR=(
AB/
PQ)
²=(
BC/
QR)
²=(
CA/
RP)
²
Pr
oofofPy
t
hagor
asTheor
em
St
at
ement
:
Asper
Py
t
hagor
ast
heor
em,
“
I
nar
i
ght
angl
edt
r
i
angl
e,t
hesum ofsquar
es
oft
wosi
desofar
i
ghtt
r
i
angl
ei
sequal
t
ot
hesquar
eoft
hehy
pot
enuseoft
het
r
i
angl
e
...
Her
e,t
hesi
deACi
st
hehy
pot
enuse,whi
chi
st
heopposi
t
eoft
her
i
ghtangl
e
...
Thesi
deABi
st
hebase,
whi
chi
st
hesi
deadj
acentt
oangl
eA
...
Pr
oof–
Consi
dert
her
i
ghtt
r
i
angl
e,
r
i
ght
angl
edatB
...
AC=AB2
……………(
i
)
Al
so,
△BDC~△ ABC
So,
CD/
BC=BC/
AC
or
,
CD
...
AC+CD
...
Title: Class 10 Triangles
Description: These notes are very helpful for class 10 students
Description: These notes are very helpful for class 10 students