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.
Statements |
Reasons |
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
Statements |
Reasons |
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
Statements |
Reasons |
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)
From Angle Side Angle to Postulates of Congruent triangle
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