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UÀÄ®§UÁð
xŠ®²y®N®Š®±
²æà gÁd±ÉÃRgï ²gÀªÁ¼ÀPÀgÀ ¸Àgï
«µÀAiÀÄ :- wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉ ªÀÄvÀÄÛ PÉÆãÀUÀ¼ÀÄ
I. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉ
II. C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ PÉÆãÀUÀ¼ÀÄ
III. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ ¹zÁÝAvÀUÀ¼ÀÄ
1. ¨ÁºÀÄ-PÉÆãÀ-¨ÁºÀÄ ¹zÁÝAvÀ (¨Á.PÉÆÃ.¨Á)
2. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ PÉÆÃ.¨Á.PÉÆÃ. ¹zÁÝAvÀ
3. ¸ÀªÀð¸ÀªÀÄvÉAiÀÄ ¨Á.¨Á.¨Á. ¹zÁÝAvÀ
4. ®A§PÉÆãÀ, PÀtð, ¨ÁºÀÄ ¹zÁÝAvÀ (®A.PÉÆÃ.¨Á)
wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉ :
MAzÀÄ wæ¨sÀÄdzÀ J¯Áè ¨ÁºÀÄUÀ¼À ªÀÄvÀÄÛ PÉÆãÀUÀ¼À C¼ÀvÉ
E£ÉÆßAzÀÄ wæ¨sÀÄdPÉÌ ¸ÀA§A¢ü¹zÀ C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ½UÉ
ªÀÄvÀÄÛ PÉÆãÀUÀ½UÉ ¸ÀªÀĪÁVzÀÝgÉ, D JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ
¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ PÉÆãÀUÀ¼ÀÄ:
¥ÀgÀ¸ÀàgÀ LPÀåªÁUÀĪÀ PÉÆãÀUÀ¼À£ÀÄß C£ÀÄgÀÆ¥À
PÉÆãÀUÀ¼ÉAzÀÄ ªÀÄvÀÄÛ ¥ÀgÀ¸ÀàgÀ LPÀåªÁUÀĪÀ ¨ÁºÀÄUÀ¼À£ÀÄß
C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÉAzÀÄ PÀgÉAiÀÄÄvÉÛêÉ.
wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ ¹zÁÝAvÀUÀ¼ÀÄ :
1) ¨ÁºÀÄ-PÉÆãÀ-¨ÁºÀÄ ¹zÁÝAvÀ (¨Á.PÉÆÃ.¨Á)
MAzÀÄ wæ¨sÀÄdzÀ JgÀqÀÄ ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ½AzÀ
GAmÁUÀĪÀ PÉÆãÀªÀÅ E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ C£ÀÄgÀÆ¥À
¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ½AzÀ GAmÁzÀ PÉÆãÀPÉÌ
¸ÀªÀĪÁVzÀÝgÉ, D JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
A
B C
D
E F
wæ¨sÀÄd ABC ªÀÄvÀÄÛ wæ¨sÀÄd DEF £À°è AB=DE,
AC=DF ªÀÄvÀÄÛ A = D DzÀÝjAzÀ ¨Á.PÉÆÃ.¨Á ¹zÁÝAvÀ¢AzÀ
ABC = DEF
2) wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ PÉÆÃ.¨Á.PÉÆÃ. ¹zÁÝAvÀ :
MAzÀÄ wæ¨sÀÄdªÀÅ JgÀqÀÄ PÉÆãÀUÀ¼ÀÄ ªÀÄvÀÄÛ
CªÀÅUÀ¼À ¸ÁªÀiÁ£Àå ¨ÁºÀÄ, E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ
JgÀqÀÄ PÉÆãÀUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ¼À ¸ÁªÀiÁ£Àå
¨ÁºÀÄUÀ½UÉ C£ÀÄgÀÆ¥ÀªÁV ¸ÀªÀÄ£ÁVzÀÝgÉ D JgÀqÀÄ
wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ
A
B C
D
E F
ABC ªÀÄvÀÄÛ DEF wæ¨sÀÄdUÀ¼À°è
B = E, C= F ªÀÄvÀÄÛ BC =EF
PÉÆÃ.¨Á.PÉÆÃ. ¹zÁÝAvÀ¢AzÀ ABC = DEF
MAzÀÄ wæ¨sÀÄdzÀ°è ¸ÀªÀĨÁºÀÄUÀ½UÉ C©üªÀÄÄRªÁVgÀĪÀ
PÉÆãÀUÀ¼ÀÄ ¸ÀªÀĪÁVgÀÄvÀÛªÉ JAzÀÄ PÀ°wzÉÝêÉ.
3) ¸ÀªÀð¸ÀªÀÄvÉAiÀÄ ¨Á.¨Á.¨Á. ¹zÁÝAvÀ :
AiÀiÁªÀÅzÉà wæ¨sÀÄdzÀ ªÀÄÆgÀÄ ¨ÁºÀÄUÀ¼ÀÄ E£ÉÆßAzÀÄ
wæ¨sÀÄdzÀ ªÀÄÆgÀÄ C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ½UÉ ¸ÀªÀÄ£ÁVzÀÝ°è D
JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
4) ®A.PÀ.¨Á. ¹zÁÝAvÀ (®A§PÉÆãÀ, PÀtð, ¨ÁºÀÄ ¹zÁÝAvÀ) :
JgÀqÀÄ ®A§PÉÆãÀ wæ¨sÀÄdUÀ¼À°è MAzÀgÀ PÀtð ªÀÄvÀÄÛ
MAzÀÄ ¨ÁºÀÄ E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ PÀtð ªÀÄvÀÄÛ C£ÀÄgÀÆ¥À
¨ÁºÀÄ«UÉ ¸ÀªÀÄ£ÁVzÁÝUÀ D ®A§PÉÆãÀ wæ¨sÀÄdUÀ¼ÀÄ
¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
A
B C
D
G E F
zÀvÀÛ : ABC ªÀÄvÀÄÛ DEF JgÀqÀÄ ®A§PÉÆãÀ wæ¨sÀÄd¼ÁVªÉ EªÀÅUÀ¼À°è
i) ABC = DEG = 900
ii) «PÀtð AC = «PÀtð DE ªÀÄvÀÄÛ
iii) AB = DE
¸ÁzsÀ¤ÃAiÀÄ : ABC = DEF
gÀZÀ£É : FE AiÀÄ£ÀÄß G ªÀgÉUÉ, EG=BC DUÀĪÀAvÉ ªÀÈ¢Þ¹ ªÀÄvÀÄÛ DG ¸ÉÃj¹
¸ÁzsÀ£É : wæ¨sÀÄd ABC ªÀÄvÀÄÛ DEG UÀ¼À£ÀÄß CªÀ¯ÉÆÃQ¹zÁUÀ
AB = DE (zÀvÁÛA±À)
BC = EG (gÀZÀ£É¬ÄAzÀ)
ABC = DEG = 900 (gÀZÀ£É¬ÄAzÀ)
¨Á.PÉÆÃ.¨Á ¹zÁÝAvÀ¢AzÀ, ABC = DEF
EzÀjAzÀ ACB = DGE ªÀÄvÀÄÛ AC=DG JAzÀÄ PÀAqÀħgÀÄvÀÛzÉ,
DzÀgÉ, AC = DF
DzÀÝjAzÀ DG = AC = DF DzÀgÉ, DGF £À°è G = F
(¸ÀªÀĨÁºÀÄUÀ½UÉ C©üªÀÄÄRªÁzÀ PÉÆãÀUÀ¼ÀÄ ¸ÀªÀĪÁVgÀÄvÀÛªÉ).
FUÀ DEF ªÀÄvÀÄÛ DEG UÀ¼À£ÀÄß ¥ÀjUÀt¹zÁUÀ, G = F
DEG = DEF EzÀjAzÀ GDE = FDE JAzÀÄ wêÀiÁð£ÀPÉÌ
§gÀ§ºÀÄzÀÄ DEG ªÀÄvÀÄÛ DEF £ÀÄß ¥ÀjUÀt¹zÁUÀ, £ÀªÀÄUÉ
DG = DF (¸Á¢ü¹zÉ)
DE = DE (G¨sÀAiÀÄ ¸ÁªÀiÁ£Àå)
DEG = DEF (¸Á¢ü¹zÉ)
¨Á.PÉÆÃ.¨Á ¹zÁÝAvÀ¢AzÀ DEG = DEF DzÀgÉ F »AzÉAiÉÄÃ
ABC = DEG JAzÀÄ ¸Á¢ü¹gÀĪÀÅzÀjAzÀ ABC = DEF DVgÀÄvÀÛzÉ.

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Module 1 traiangle

  • 1. UÀÄ®§UÁð xŠ®²y®N®Š®± ²æà gÁd±ÉÃRgï ²gÀªÁ¼ÀPÀgÀ ¸Àgï «µÀAiÀÄ :- wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉ ªÀÄvÀÄÛ PÉÆãÀUÀ¼ÀÄ
  • 2. I. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉ II. C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ PÉÆãÀUÀ¼ÀÄ III. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ ¹zÁÝAvÀUÀ¼ÀÄ 1. ¨ÁºÀÄ-PÉÆãÀ-¨ÁºÀÄ ¹zÁÝAvÀ (¨Á.PÉÆÃ.¨Á) 2. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ PÉÆÃ.¨Á.PÉÆÃ. ¹zÁÝAvÀ 3. ¸ÀªÀð¸ÀªÀÄvÉAiÀÄ ¨Á.¨Á.¨Á. ¹zÁÝAvÀ 4. ®A§PÉÆãÀ, PÀtð, ¨ÁºÀÄ ¹zÁÝAvÀ (®A.PÉÆÃ.¨Á)
  • 3. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉ : MAzÀÄ wæ¨sÀÄdzÀ J¯Áè ¨ÁºÀÄUÀ¼À ªÀÄvÀÄÛ PÉÆãÀUÀ¼À C¼ÀvÉ E£ÉÆßAzÀÄ wæ¨sÀÄdPÉÌ ¸ÀA§A¢ü¹zÀ C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ½UÉ ªÀÄvÀÄÛ PÉÆãÀUÀ½UÉ ¸ÀªÀĪÁVzÀÝgÉ, D JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
  • 4. C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ PÉÆãÀUÀ¼ÀÄ: ¥ÀgÀ¸ÀàgÀ LPÀåªÁUÀĪÀ PÉÆãÀUÀ¼À£ÀÄß C£ÀÄgÀÆ¥À PÉÆãÀUÀ¼ÉAzÀÄ ªÀÄvÀÄÛ ¥ÀgÀ¸ÀàgÀ LPÀåªÁUÀĪÀ ¨ÁºÀÄUÀ¼À£ÀÄß C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÉAzÀÄ PÀgÉAiÀÄÄvÉÛêÉ.
  • 5. wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ ¹zÁÝAvÀUÀ¼ÀÄ : 1) ¨ÁºÀÄ-PÉÆãÀ-¨ÁºÀÄ ¹zÁÝAvÀ (¨Á.PÉÆÃ.¨Á) MAzÀÄ wæ¨sÀÄdzÀ JgÀqÀÄ ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ½AzÀ GAmÁUÀĪÀ PÉÆãÀªÀÅ E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ½AzÀ GAmÁzÀ PÉÆãÀPÉÌ ¸ÀªÀĪÁVzÀÝgÉ, D JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
  • 6. A B C D E F wæ¨sÀÄd ABC ªÀÄvÀÄÛ wæ¨sÀÄd DEF £À°è AB=DE, AC=DF ªÀÄvÀÄÛ A = D DzÀÝjAzÀ ¨Á.PÉÆÃ.¨Á ¹zÁÝAvÀ¢AzÀ ABC = DEF
  • 7. 2) wæ¨sÀÄdUÀ¼À ¸ÀªÀð¸ÀªÀÄvÉUÉ PÉÆÃ.¨Á.PÉÆÃ. ¹zÁÝAvÀ : MAzÀÄ wæ¨sÀÄdªÀÅ JgÀqÀÄ PÉÆãÀUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ¼À ¸ÁªÀiÁ£Àå ¨ÁºÀÄ, E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ JgÀqÀÄ PÉÆãÀUÀ¼ÀÄ ªÀÄvÀÄÛ CªÀÅUÀ¼À ¸ÁªÀiÁ£Àå ¨ÁºÀÄUÀ½UÉ C£ÀÄgÀÆ¥ÀªÁV ¸ÀªÀÄ£ÁVzÀÝgÉ D JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ
  • 8. A B C D E F ABC ªÀÄvÀÄÛ DEF wæ¨sÀÄdUÀ¼À°è B = E, C= F ªÀÄvÀÄÛ BC =EF PÉÆÃ.¨Á.PÉÆÃ. ¹zÁÝAvÀ¢AzÀ ABC = DEF MAzÀÄ wæ¨sÀÄdzÀ°è ¸ÀªÀĨÁºÀÄUÀ½UÉ C©üªÀÄÄRªÁVgÀĪÀ PÉÆãÀUÀ¼ÀÄ ¸ÀªÀĪÁVgÀÄvÀÛªÉ JAzÀÄ PÀ°wzÉÝêÉ.
  • 9. 3) ¸ÀªÀð¸ÀªÀÄvÉAiÀÄ ¨Á.¨Á.¨Á. ¹zÁÝAvÀ : AiÀiÁªÀÅzÉà wæ¨sÀÄdzÀ ªÀÄÆgÀÄ ¨ÁºÀÄUÀ¼ÀÄ E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ ªÀÄÆgÀÄ C£ÀÄgÀÆ¥À ¨ÁºÀÄUÀ½UÉ ¸ÀªÀÄ£ÁVzÀÝ°è D JgÀqÀÄ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
  • 10. 4) ®A.PÀ.¨Á. ¹zÁÝAvÀ (®A§PÉÆãÀ, PÀtð, ¨ÁºÀÄ ¹zÁÝAvÀ) : JgÀqÀÄ ®A§PÉÆãÀ wæ¨sÀÄdUÀ¼À°è MAzÀgÀ PÀtð ªÀÄvÀÄÛ MAzÀÄ ¨ÁºÀÄ E£ÉÆßAzÀÄ wæ¨sÀÄdzÀ PÀtð ªÀÄvÀÄÛ C£ÀÄgÀÆ¥À ¨ÁºÀÄ«UÉ ¸ÀªÀÄ£ÁVzÁÝUÀ D ®A§PÉÆãÀ wæ¨sÀÄdUÀ¼ÀÄ ¸ÀªÀð¸ÀªÀĪÁVgÀÄvÀÛªÉ.
  • 11. A B C D G E F zÀvÀÛ : ABC ªÀÄvÀÄÛ DEF JgÀqÀÄ ®A§PÉÆãÀ wæ¨sÀÄd¼ÁVªÉ EªÀÅUÀ¼À°è i) ABC = DEG = 900 ii) «PÀtð AC = «PÀtð DE ªÀÄvÀÄÛ iii) AB = DE ¸ÁzsÀ¤ÃAiÀÄ : ABC = DEF gÀZÀ£É : FE AiÀÄ£ÀÄß G ªÀgÉUÉ, EG=BC DUÀĪÀAvÉ ªÀÈ¢Þ¹ ªÀÄvÀÄÛ DG ¸ÉÃj¹ ¸ÁzsÀ£É : wæ¨sÀÄd ABC ªÀÄvÀÄÛ DEG UÀ¼À£ÀÄß CªÀ¯ÉÆÃQ¹zÁUÀ AB = DE (zÀvÁÛA±À) BC = EG (gÀZÀ£É¬ÄAzÀ) ABC = DEG = 900 (gÀZÀ£É¬ÄAzÀ)
  • 12. ¨Á.PÉÆÃ.¨Á ¹zÁÝAvÀ¢AzÀ, ABC = DEF EzÀjAzÀ ACB = DGE ªÀÄvÀÄÛ AC=DG JAzÀÄ PÀAqÀħgÀÄvÀÛzÉ, DzÀgÉ, AC = DF DzÀÝjAzÀ DG = AC = DF DzÀgÉ, DGF £À°è G = F (¸ÀªÀĨÁºÀÄUÀ½UÉ C©üªÀÄÄRªÁzÀ PÉÆãÀUÀ¼ÀÄ ¸ÀªÀĪÁVgÀÄvÀÛªÉ). FUÀ DEF ªÀÄvÀÄÛ DEG UÀ¼À£ÀÄß ¥ÀjUÀt¹zÁUÀ, G = F DEG = DEF EzÀjAzÀ GDE = FDE JAzÀÄ wêÀiÁð£ÀPÉÌ §gÀ§ºÀÄzÀÄ DEG ªÀÄvÀÄÛ DEF £ÀÄß ¥ÀjUÀt¹zÁUÀ, £ÀªÀÄUÉ DG = DF (¸Á¢ü¹zÉ) DE = DE (G¨sÀAiÀÄ ¸ÁªÀiÁ£Àå) DEG = DEF (¸Á¢ü¹zÉ) ¨Á.PÉÆÃ.¨Á ¹zÁÝAvÀ¢AzÀ DEG = DEF DzÀgÉ F »AzÉAiÉÄà ABC = DEG JAzÀÄ ¸Á¢ü¹gÀĪÀÅzÀjAzÀ ABC = DEF DVgÀÄvÀÛzÉ.