Criteria for Similarity of Triangles
Friends, aaj ke iss article mein hum janenge triangles ke similarity ke different criteria ke baare mein. Iss article mein hum ye seekhenge ki kab do(two) ya do(two) se jyada triangles aapas mein similar hote hain aur wo kaun si techniques/methods hain jisse hum ye pata kar sake ki diye gaye do(two) ya do(two) se jyada triangles similar hain.
Friends mera naam hai Dheeraj Sahni aur aap hain iss waqt meri website www.mathshindi.com par. Bane rahiye mere saath iss article ke last tak aur increase kijiye apni knowledge ko....
Friends aaj ka hamara topic hai–Criteria for Similarity of Triangles
Wo kaun se methods/techniques hain jinko apply karke diye gaye do(two) ya do(two) se jyada triangles ko similar prove kar sakte hain– ye jaanne se pahle hame ye jaanna hoga ki similar triangles ka matlab kya hota hai.
Similar triangles
Koi bhi diye gaye do(two) triangles tab similar kahe jaate hain jab unke:
(i) Corresponding angles equal ho
(ii) Corresponding sides same ratio(proportion) mein ho
Matlab, maan lijiye do two triangles ΔABC aur ΔDEF diye hue hain aur agar
(i) ∠A = ∠D, ∠B = ∠E, ∠C = ∠F aur
(ii) AB = BC = CA
DE EF FD
ho to dono triangles ΔABC aur ΔDEF similar kahe jaayenge (diagram dekhiye).
[Note: Corresponding angles ka matlab hota hai dono triangles mein wo angles jo same jagah/place par aate hain, jaise ki triangle ΔABC mein angle ∠A ko lijiye. Ab triangle ΔDEF ko dekhiye ΔDEF mein ∠A ki jagah/place par ∠D hai. So, ∠A aur ∠D ek doosre ke corresponding angle hue. And ∠B aur ∠E, ∠C aur ∠F ek doosre ke corresponding angle hue. Isi tarah se corresponding sides bhi hoti hai AB aur DE, AC aur DF, BC aur EF ek doosre ke corresponding sides hain].
Agar koi two triangles similar hai to hum is fact ko ‘∼’ ke sign se show karte hain. Sign ‘∼’ ka matlab hota hai– ‘is similar to’. Example ke liye abhi jinn do(two) triangles ΔABC aur ΔDEF ka maine example diya wo dono triangles similar hai. Matlab ΔABC is similar to ΔDEF. Iss statement ko aise likhte hain– ΔABC∼ΔDEF.
Kya do(two) triangles ki similarity check karne ke liye hame hamesha(always) saare(all) equality relations (3 corresponding angles and 3 ratio of corresponding sides) pata hona chahiye? Iss sawaal ka jawab hai– Nahi! Kisi bhi do(two) triangles ki similarity check karne ke liye hame sare(all) equality relations pata karne ki jaroorat nahi hai hame sirf kuch hi (minimum three) equality relation pata honi chahiye aur sirf inn informations ke use se hi hum diye gaye do(two) triangles ko similar prove kar sakte hain.
Friends, ab me baat karne jaa raha hoon kuch methods/techniques ki jiske use se hum ye pata kar sakte hain ki diye gaye do(two) triangles similar hai ya nahi. Actually ye koi methods ya techniques nahi hai; yeh kuch theorems hai jiska use do(two) ya do(two) se jyada triangles ki similarity check ki jaati hai. To chaliye ek-ek karke jante hain inn theorems ke baare mein–
1. AAA similarity criterion
“If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.
[Agar diye gaye do(two) triangles mein, corresponding angles equal ho to unn triangles ki corresponding sides bhi same ratio (proportion) mein hoti hai aur iss tarah wo dono triangles similar ho jaate hain].
AAA similarity criterion ka poora naam hai– Angle-Angle-Angle similarity criterion.
Proof of AAA similarity criterion
Given: Do(two) triangles ΔABC aur ΔDEF jisme ∠A = ∠D, ∠B = ∠E aur ∠C = ∠F hai.
To prove: ΔABC∼ΔDEF
Construction: Cut kijiye DP = AB aur DQ = AC aur PQ ko join kar dijiye.
Proof: Ab ΔABC ≅ ΔDPQ hua.
Isse hame ye milta hai, ∠B = ∠P = ∠E
aur PQ∥EF.
Isiliye, DP = DQ (Using BPT)
PE QF
Isiliye, AB = AC
DE DF
Isi tarah se AB = BC aur isiliye
DE EF
AB = BC = AC
DE EF DF
Iska matlab ye hua ki ΔABC∼ΔDEF.
[Agar aapko nahi pata ki BPT kya hai to ye jaanne ke liye iss par click kijiye »»»
Basic Proportionality Theorem (Thales Theorem)].
Read also:
What is quadratic equation ?
Pair of Linear Equations in Two Variables
Arithmetic Progression (AP)
2. AA similarity criterion
“If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar”.
[Agar diye gaye do(two) triangles ke koi bhi do(two) corresponding angles equal ho to wo dono triangles similar hote hain].
AA similarity criterion ka poora naam hai– Angle-Angle similarity criterion.
Proof of AA similarity criterion
Iska proof same to same AAA similarity criterion ke proof ke equivalent hai kyunki agar kisi do(two) triangles ke koi do(two) coressponding angle equal ho to dono triangles ka teesra(third) angle bhi equal hoga. Iss tarah dono triangles ke teeno(three) angles equal honge aur phir iska prove same to same AAA similarity criterion ki tarah diya jaa sakta hai.
3. SSS similarity criterion
“If in two triangles, sides of one triangle are proportional to ( i.e., in the same ratio) of the side of the other triangle, then their corresponding angles are equal and hence the two triangles are similar”.
[Agar diye gaye do(two) triangles mein ek triangle ki sides, doosre triangle ke sides ke same ratio (proportion) mein ho to unke corresponding angles bhi equal hote hain aur iss tarah wo do triangles similar hote hain].
SSS similarity criterion ka poora naam hai– Side-Side-Side similarity criterion.
Proof of SSS similarity criterion
Given: Two triangles ΔABC aur ΔDEF jiss mein
AB = BC = CA
DE EF FD
To prove: ΔABC∼ΔDEF
Construction: Cut kijiye DP = KB aur DQ = AC aur PQ ko join kar dijiye.
Proof: AB = AC (given) ---------(1)
DE DF
DP = DQ (AB = DP and AC = DQ)
DE DF
So, PQ∥EF (By converse of BPT)
So, ∠DPQ = ∠E and ∠DQP =∠F (Corresponding angles)
So, ΔDPQ∼ΔDEF (By AA similarity criterion) -----------(2)
So, DP = PQ -----------(3)
DE EF
Hence, AB = PQ (DP = AB) -----------(4)
DE EF
Hence, PQ = BC [From (1), (2), and (4)]
EF EF
So, PQ = BC
So, ΔABC ≅ ΔDPQ (By SSS congruency) -----(5)
So, ΔABC∼ΔDEF [From (2) and (5)]
4. SAS similarity criterion
“If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar”.
[Agar diye gaye do(two) triangles mein dono triangles ke ek-ek angle equal ho aur uss angle ko include karne waali sides same ratio mein (proportional) ho to wo dono triangles similar hote hai].
SAS similarity criterion ka poora naam hai– Side-Angle-Side similarity criterion.
Proof of SAS similarity criterion
Given: Do(two) triangles ΔABC aur ΔDEF jisme
AB = AC aur ∠A = ∠D diya
DE DF
hua hai.
To prove: ΔABC∼ΔDEF
Construction: Cut kijiye DP = AB, DQ = AC aur join kar dijiye PQ ko
Proof: Now, AB = DP (By construction)
∠A = ∠D (Given)
AC = DQ (By construction) So, ΔABC ≅ ΔDPQ (By SAS cogruency rule)
Now, AB = AC (Given)
DE DF
DP = DQ
DE DF
So, PQ∥EF (By converse of BPT)
∠DPQ = ∠E and ∠DQP = ∠F (Corresponding angles)
So, ΔDPQ∼ΔDEF (By AAA similarity criterion)
So, ΔABC∼ΔDEF.
I hope ki mera ye article aap ko pasand aaya hoga. Agar aap ko mera ye article pasand aaya ho to comment karke hame bata sakte hain.
THANKS FOR READING THIS BLOG.
Friends mera naam hai Dheeraj Sahni aur aap hain iss waqt meri website www.mathshindi.com par. Bane rahiye mere saath iss article ke last tak aur increase kijiye apni knowledge ko....
Friends aaj ka hamara topic hai–Criteria for Similarity of Triangles
Wo kaun se methods/techniques hain jinko apply karke diye gaye do(two) ya do(two) se jyada triangles ko similar prove kar sakte hain– ye jaanne se pahle hame ye jaanna hoga ki similar triangles ka matlab kya hota hai.
Similar triangles
Koi bhi diye gaye do(two) triangles tab similar kahe jaate hain jab unke:
(i) Corresponding angles equal ho
(ii) Corresponding sides same ratio(proportion) mein ho
Matlab, maan lijiye do two triangles ΔABC aur ΔDEF diye hue hain aur agar
(i) ∠A = ∠D, ∠B = ∠E, ∠C = ∠F aur
(ii) AB = BC = CA
DE EF FD
ho to dono triangles ΔABC aur ΔDEF similar kahe jaayenge (diagram dekhiye).
[Note: Corresponding angles ka matlab hota hai dono triangles mein wo angles jo same jagah/place par aate hain, jaise ki triangle ΔABC mein angle ∠A ko lijiye. Ab triangle ΔDEF ko dekhiye ΔDEF mein ∠A ki jagah/place par ∠D hai. So, ∠A aur ∠D ek doosre ke corresponding angle hue. And ∠B aur ∠E, ∠C aur ∠F ek doosre ke corresponding angle hue. Isi tarah se corresponding sides bhi hoti hai AB aur DE, AC aur DF, BC aur EF ek doosre ke corresponding sides hain].
Agar koi two triangles similar hai to hum is fact ko ‘∼’ ke sign se show karte hain. Sign ‘∼’ ka matlab hota hai– ‘is similar to’. Example ke liye abhi jinn do(two) triangles ΔABC aur ΔDEF ka maine example diya wo dono triangles similar hai. Matlab ΔABC is similar to ΔDEF. Iss statement ko aise likhte hain– ΔABC∼ΔDEF.
Kya do(two) triangles ki similarity check karne ke liye hame hamesha(always) saare(all) equality relations (3 corresponding angles and 3 ratio of corresponding sides) pata hona chahiye? Iss sawaal ka jawab hai– Nahi! Kisi bhi do(two) triangles ki similarity check karne ke liye hame sare(all) equality relations pata karne ki jaroorat nahi hai hame sirf kuch hi (minimum three) equality relation pata honi chahiye aur sirf inn informations ke use se hi hum diye gaye do(two) triangles ko similar prove kar sakte hain.
Friends, ab me baat karne jaa raha hoon kuch methods/techniques ki jiske use se hum ye pata kar sakte hain ki diye gaye do(two) triangles similar hai ya nahi. Actually ye koi methods ya techniques nahi hai; yeh kuch theorems hai jiska use do(two) ya do(two) se jyada triangles ki similarity check ki jaati hai. To chaliye ek-ek karke jante hain inn theorems ke baare mein–
1. AAA similarity criterion
“If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.
[Agar diye gaye do(two) triangles mein, corresponding angles equal ho to unn triangles ki corresponding sides bhi same ratio (proportion) mein hoti hai aur iss tarah wo dono triangles similar ho jaate hain].
AAA similarity criterion ka poora naam hai– Angle-Angle-Angle similarity criterion.
Proof of AAA similarity criterion
Given: Do(two) triangles ΔABC aur ΔDEF jisme ∠A = ∠D, ∠B = ∠E aur ∠C = ∠F hai.
To prove: ΔABC∼ΔDEF
Construction: Cut kijiye DP = AB aur DQ = AC aur PQ ko join kar dijiye.
Proof: Ab ΔABC ≅ ΔDPQ hua.
Isse hame ye milta hai, ∠B = ∠P = ∠E
aur PQ∥EF.
Isiliye, DP = DQ (Using BPT)
PE QF
Isiliye, AB = AC
DE DF
Isi tarah se AB = BC aur isiliye
DE EF
AB = BC = AC
DE EF DF
Iska matlab ye hua ki ΔABC∼ΔDEF.
[Agar aapko nahi pata ki BPT kya hai to ye jaanne ke liye iss par click kijiye »»»
Basic Proportionality Theorem (Thales Theorem)].
Read also:
2. AA similarity criterion
“If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar”.
[Agar diye gaye do(two) triangles ke koi bhi do(two) corresponding angles equal ho to wo dono triangles similar hote hain].
AA similarity criterion ka poora naam hai– Angle-Angle similarity criterion.
Proof of AA similarity criterion
Iska proof same to same AAA similarity criterion ke proof ke equivalent hai kyunki agar kisi do(two) triangles ke koi do(two) coressponding angle equal ho to dono triangles ka teesra(third) angle bhi equal hoga. Iss tarah dono triangles ke teeno(three) angles equal honge aur phir iska prove same to same AAA similarity criterion ki tarah diya jaa sakta hai.
3. SSS similarity criterion
“If in two triangles, sides of one triangle are proportional to ( i.e., in the same ratio) of the side of the other triangle, then their corresponding angles are equal and hence the two triangles are similar”.
[Agar diye gaye do(two) triangles mein ek triangle ki sides, doosre triangle ke sides ke same ratio (proportion) mein ho to unke corresponding angles bhi equal hote hain aur iss tarah wo do triangles similar hote hain].
SSS similarity criterion ka poora naam hai– Side-Side-Side similarity criterion.
Proof of SSS similarity criterion
Given: Two triangles ΔABC aur ΔDEF jiss mein
AB = BC = CA
DE EF FD
To prove: ΔABC∼ΔDEF
Construction: Cut kijiye DP = KB aur DQ = AC aur PQ ko join kar dijiye.
Proof: AB = AC (given) ---------(1)
DE DF
DP = DQ (AB = DP and AC = DQ)
DE DF
So, PQ∥EF (By converse of BPT)
So, ∠DPQ = ∠E and ∠DQP =∠F (Corresponding angles)
So, ΔDPQ∼ΔDEF (By AA similarity criterion) -----------(2)
So, DP = PQ -----------(3)
DE EF
Hence, AB = PQ (DP = AB) -----------(4)
DE EF
Hence, PQ = BC [From (1), (2), and (4)]
EF EF
So, PQ = BC
So, ΔABC ≅ ΔDPQ (By SSS congruency) -----(5)
So, ΔABC∼ΔDEF [From (2) and (5)]
4. SAS similarity criterion
“If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar”.
[Agar diye gaye do(two) triangles mein dono triangles ke ek-ek angle equal ho aur uss angle ko include karne waali sides same ratio mein (proportional) ho to wo dono triangles similar hote hai].
SAS similarity criterion ka poora naam hai– Side-Angle-Side similarity criterion.
Proof of SAS similarity criterion
Given: Do(two) triangles ΔABC aur ΔDEF jisme
AB = AC aur ∠A = ∠D diya
DE DF
hua hai.
To prove: ΔABC∼ΔDEF
Construction: Cut kijiye DP = AB, DQ = AC aur join kar dijiye PQ ko
Proof: Now, AB = DP (By construction)
∠A = ∠D (Given)
AC = DQ (By construction) So, ΔABC ≅ ΔDPQ (By SAS cogruency rule)
Now, AB = AC (Given)
DE DF
DP = DQ
DE DF
So, PQ∥EF (By converse of BPT)
∠DPQ = ∠E and ∠DQP = ∠F (Corresponding angles)
So, ΔDPQ∼ΔDEF (By AAA similarity criterion)
So, ΔABC∼ΔDEF.
I hope ki mera ye article aap ko pasand aaya hoga. Agar aap ko mera ye article pasand aaya ho to comment karke hame bata sakte hain.
THANKS FOR READING THIS BLOG.
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