CLASS-10th 1.5tMILARITY GEOMETRY.
Lel's Stucdy:
Ratio of the asveas
D
tuwo trian gles and selated propextles [-1
Basic Paepoubionality thearem [BPTJ [12]
Conve se
obasic Propontionality theozem [12]
Preperty oan angle
0 bisector of
triangle [1:2]
a
Convexse oproperty
an
D angle bisector e triangle [13 a
Preperty three parallel ines and their
transversals [1-2
Similarity e triangles.[13]
Tes eSirni larity
e triangles [4:5]
Theorem areas Similar
triangles.[14]
PRACTICE SET +PROBLem
|1:1, 1:2,1:3,1:4 SET 1
NOTES BY: GALAxY PARMAR
, ReCALL Conditiem 1: I tha huights et toth triangles
equal then-
ae
1f we say numbers 'a and b axe io Heights P
Cwritten uatio then it is nACAABC)- BC XAD BC Xh A
as a b or
=
Same
PACAPQRR) RXPSs
e a ef a triangle = Base X BC= bi
height R ba/
Basic Concept of Practice set 1:1
ACAABC)- 61
Ratio e aveas o two tviangles. ACAPQR) b2 b-
D
R
h e Hatio o the PROPERTY he Hatie e the
axeas e two triangles is equal to the aueas e tws triang! es wit
SAatis ethe equal heioht
preductsthein bases and 9 ual to the Hatio o their
heights coves ponding Coresponcbng bases
Conditien 2: t the bases e both triangles ave equal
then-
A
he CAAEBC). ABX CD X h =h
A CAABP)
BasesE
ABX P bXha Same
B
bi-
& ACAABC)= hi
R A
ACDABC) BC XAD ACAABD
ACAPOR QR X PS
A CAABC.
RoPERTY: he Hatio o the aseos tws
bi Xhl buianglus
equal bases is equal ts tha eatio uih
A CAPOR) b2X ha ot theie
Ccoepending hwights.
Hence, he uatie oaHeas 6uso tiangles Con dition 3: I the bases and
=
biXh aa Raual then-
highs both triangles
ba Kh2 Bases and
PROPERTY: heighis sand
Now en side the
ollwing Conditiony. he Aakie the
aueas e tuwo
equal bases and egual triangles with
1 OR
hetghts eqial to
ea e we triangles an e
equak.
, asic Concept of Practice set 1:2 (] Property of an
a triangle
ongle bisectay of [5) Kopesty of thrne pevoll)
alBasic Prepevtionality th esrem ines and theiv transve rsal
statenent: he bisectos e Statement He natio of the
[BPT] aungle o a triangle divicdes he interceptS made on a tanSveua
Statement: I{ a line payallel to a side ea side opposite to the angle in by three pauallel Lines is equal te
riange intexsects the uemaining sidu in twe the ratio o the xemaini the ratio e the CoTres
distinct psint S, then the line aivides the &ides. pending
sides in the Same A intencepts made en any othur
propsatiem. In AABC
b
In AABC, ä tisecto o transvexsad by the Same parallel
a y CE LACB
ines
T Une Il side BC thun AE= AC
EB BC
then A AQ=
PD QC
] Converse o angle bisector theerem
In AABC, M
Converse of Basic proportionality Peint Eon side AB
Such thcut.
theorem
Statement 8 Tf a line divide any tuo sides
AEB Ac
E AC
BC
then ay CE bisect
a
triangle "in the Same ratio, then the Lne II line m l| Lne n
Line is paua lel to the third side. 2 ACB
and ti and t2 is transvets als
In AABC, then AB -P
QR
AP
PP
-A
than ine ll sideBC
, asic Cencept of Practice set 1:3 0 AAAtest: Jor a given one-to-one (2) SAS test: Ta a oiven one-ts
CoTresponden ce between the vertice ome coespendence between the vertices
imilay Fiqu res : iqunes which ane o triangles , i the Covres pending otwo Eiangleu, if two sides o ene
8ame
Shape butmay di
be e
otwo
fewent Sizes aue cauled
angles ane con q'vuent, then the two aingle. are poportional to the
Similay figuwres. Cormes pond ing sides o the othe
triangles ane Similar triande
ond the ongfe include d bs them are
Simila7 triangles: Yon a
given ene to one
Conqzuent, then the twe trangles
are similar.
COHHes pend en ce between the vevtices
tw
triangle, B60
50
AA
Ci) their CoTespen ding angles aue Cengsuent and
In kig, unde the
Cii) their Corvepending Sids Me in prspor tion, couspen dence,
A BC PQ R
then the twe triangles ane said ts be
LA LP, 2B S29, 2c 2R
Similay triangles.
hen by AAA teut n a , undut the coeutespendence
AABC A PQR PgR XYZ,
P 9R - and
AAtest Jor a given ene-to-9ne Comespo XY
B Ondente between the veuticeu o two trangles, Lg >LY
t u o angles o one triangte" ane cenqsuent hun by SAS teut, APGR NAXY2
In AABC and APGR
with the corespen ding twe anglu the
LA LP, 2BLQ,Lc L R and
other triangle, then the two
o (3)SSStest: T three sides o
AB- BC Similar. tianglu are ene triangle axe preportional tothe
P9 R PR
AA
thoee CoTejponding sidea the
then AABC and APQ R ae simila 0ther triangle, then the twe
triangle are Simila.
ie AABC APQR
In g und et the
CoLespondence 27
Tests for similo7ity triangles: 8
hee aHe three tests toa similanty o triangles LMN RST,
M 2s, zNLT Jn
tig, undey the cOHespond ence
(DAAA test o similavity or AA tet oSimi ABc GEF AB =BC -Ac -S
lanty hen bu AA Lest, GE EF GF
a)SAS test o Similarity ALMN ARST hen by SSS test ARBC vA
(a) SSS testof similarity GEF
Lel's Stucdy:
Ratio of the asveas
D
tuwo trian gles and selated propextles [-1
Basic Paepoubionality thearem [BPTJ [12]
Conve se
obasic Propontionality theozem [12]
Preperty oan angle
0 bisector of
triangle [1:2]
a
Convexse oproperty
an
D angle bisector e triangle [13 a
Preperty three parallel ines and their
transversals [1-2
Similarity e triangles.[13]
Tes eSirni larity
e triangles [4:5]
Theorem areas Similar
triangles.[14]
PRACTICE SET +PROBLem
|1:1, 1:2,1:3,1:4 SET 1
NOTES BY: GALAxY PARMAR
, ReCALL Conditiem 1: I tha huights et toth triangles
equal then-
ae
1f we say numbers 'a and b axe io Heights P
Cwritten uatio then it is nACAABC)- BC XAD BC Xh A
as a b or
=
Same
PACAPQRR) RXPSs
e a ef a triangle = Base X BC= bi
height R ba/
Basic Concept of Practice set 1:1
ACAABC)- 61
Ratio e aveas o two tviangles. ACAPQR) b2 b-
D
R
h e Hatio o the PROPERTY he Hatie e the
axeas e two triangles is equal to the aueas e tws triang! es wit
SAatis ethe equal heioht
preductsthein bases and 9 ual to the Hatio o their
heights coves ponding Coresponcbng bases
Conditien 2: t the bases e both triangles ave equal
then-
A
he CAAEBC). ABX CD X h =h
A CAABP)
BasesE
ABX P bXha Same
B
bi-
& ACAABC)= hi
R A
ACDABC) BC XAD ACAABD
ACAPOR QR X PS
A CAABC.
RoPERTY: he Hatio o the aseos tws
bi Xhl buianglus
equal bases is equal ts tha eatio uih
A CAPOR) b2X ha ot theie
Ccoepending hwights.
Hence, he uatie oaHeas 6uso tiangles Con dition 3: I the bases and
=
biXh aa Raual then-
highs both triangles
ba Kh2 Bases and
PROPERTY: heighis sand
Now en side the
ollwing Conditiony. he Aakie the
aueas e tuwo
equal bases and egual triangles with
1 OR
hetghts eqial to
ea e we triangles an e
equak.
, asic Concept of Practice set 1:2 (] Property of an
a triangle
ongle bisectay of [5) Kopesty of thrne pevoll)
alBasic Prepevtionality th esrem ines and theiv transve rsal
statenent: he bisectos e Statement He natio of the
[BPT] aungle o a triangle divicdes he interceptS made on a tanSveua
Statement: I{ a line payallel to a side ea side opposite to the angle in by three pauallel Lines is equal te
riange intexsects the uemaining sidu in twe the ratio o the xemaini the ratio e the CoTres
distinct psint S, then the line aivides the &ides. pending
sides in the Same A intencepts made en any othur
propsatiem. In AABC
b
In AABC, ä tisecto o transvexsad by the Same parallel
a y CE LACB
ines
T Une Il side BC thun AE= AC
EB BC
then A AQ=
PD QC
] Converse o angle bisector theerem
In AABC, M
Converse of Basic proportionality Peint Eon side AB
Such thcut.
theorem
Statement 8 Tf a line divide any tuo sides
AEB Ac
E AC
BC
then ay CE bisect
a
triangle "in the Same ratio, then the Lne II line m l| Lne n
Line is paua lel to the third side. 2 ACB
and ti and t2 is transvets als
In AABC, then AB -P
QR
AP
PP
-A
than ine ll sideBC
, asic Cencept of Practice set 1:3 0 AAAtest: Jor a given one-to-one (2) SAS test: Ta a oiven one-ts
CoTresponden ce between the vertice ome coespendence between the vertices
imilay Fiqu res : iqunes which ane o triangles , i the Covres pending otwo Eiangleu, if two sides o ene
8ame
Shape butmay di
be e
otwo
fewent Sizes aue cauled
angles ane con q'vuent, then the two aingle. are poportional to the
Similay figuwres. Cormes pond ing sides o the othe
triangles ane Similar triande
ond the ongfe include d bs them are
Simila7 triangles: Yon a
given ene to one
Conqzuent, then the twe trangles
are similar.
COHHes pend en ce between the vevtices
tw
triangle, B60
50
AA
Ci) their CoTespen ding angles aue Cengsuent and
In kig, unde the
Cii) their Corvepending Sids Me in prspor tion, couspen dence,
A BC PQ R
then the twe triangles ane said ts be
LA LP, 2B S29, 2c 2R
Similay triangles.
hen by AAA teut n a , undut the coeutespendence
AABC A PQR PgR XYZ,
P 9R - and
AAtest Jor a given ene-to-9ne Comespo XY
B Ondente between the veuticeu o two trangles, Lg >LY
t u o angles o one triangte" ane cenqsuent hun by SAS teut, APGR NAXY2
In AABC and APGR
with the corespen ding twe anglu the
LA LP, 2BLQ,Lc L R and
other triangle, then the two
o (3)SSStest: T three sides o
AB- BC Similar. tianglu are ene triangle axe preportional tothe
P9 R PR
AA
thoee CoTejponding sidea the
then AABC and APQ R ae simila 0ther triangle, then the twe
triangle are Simila.
ie AABC APQR
In g und et the
CoLespondence 27
Tests for similo7ity triangles: 8
hee aHe three tests toa similanty o triangles LMN RST,
M 2s, zNLT Jn
tig, undey the cOHespond ence
(DAAA test o similavity or AA tet oSimi ABc GEF AB =BC -Ac -S
lanty hen bu AA Lest, GE EF GF
a)SAS test o Similarity ALMN ARST hen by SSS test ARBC vA
(a) SSS testof similarity GEF
