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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