Angle Side Angle Postulate

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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

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• 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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