Angle Side Angle Postulate
Angle side angle postulate (ASA) - > If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent by angle-side-angle postulate.
∠B ≅ ∠E, BC ≅ EF and ∠C ≅ ∠F
∴ ΔABC ≅ Δ DEF by ASA
Examples :
1)
Given : ∠BAC = ∠DAC and ∠BCA = ∠DCA
Prove that : AB = AD and CB = CD.
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| 1) ∠BAC = ∠DAC | 1) Given |
| 2) AC = AC | 2) Reflexive |
| 3) ∠BCA = ∠DCA | 3) Given |
| 4) ΔBAC ≅ ΔDAC | 4) By ASA (angle side angle postulate) |
| 5) AB = AD | 5) CPCTC |
| 6) CB = CD | 6) CPCTC |
2)
Given : ∠BCD = ∠ADC and ∠ACB = ∠BDA
Prove that : AD = BC and ∠A = ∠B
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| 1) ∠BCD = ∠ADC | 1) Given |
| 2) ∠ACB = ∠BDA | 2) Given |
| 3) ∠BCD + ∠ACB = ∠ADC + ∠BDA | 3) Adding (1) and (2) |
| 4) ∠ACD = ∠BDC | 4) Addition property |
| 5) CD = CD | 5) Reflexive |
| 6) ΔACD ≅ ΔBDC | 6) By ASA postulate |
| 7) AD = BC | 7) CPCTC |
| 8) ∠A = ∠B | 8) CPCTC |
3)
Given : AC = BC , ∠DCA = ∠ECB and ∠DBC = ∠EAC
Prove that : i) ΔDBC ≅ ΔEAC
(ii) DC = EC and (iii) BD = AE
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| 1) ∠DCA = ∠ECB | 1) Given |
| 2) ∠DCA + ∠ECD = ∠ECB + ∠ECD | 2) Adding angle ∠ECD both sides in (1) |
| 3) ∠ECA = ∠DCB | 3) Addition property |
| 4) BC = AC | 4) Given |
| 5) ∠DBC = ∠EAC | 5) Given |
| 6) ΔDBC ≅ ΔEAC | 6) By ASA postulate |
| 7) DC = EC | 7) CPCTC |
| 8) BD = AE | 8) CPCTC |
• Side Angle Side Postulate
• Side Side Side Postulate
• Angle Angle Side Postulate
• Angle Side Angle Postulate
• HL postulate(Hypotenuse – Leg OR RHS)
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