Angle Angle Side Postulate
Angle angle side postulate (AAS) > If two angles and a nonincluded side of a triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent by angle angle side postulate.
∠B ≅ ∠E , ∠C ≅ ∠F and AC ≅ DF
∴ Δ ABC ≅ Δ DEF by AAS
Theorem 1 : If two angles of a triangle are equal, then sides opposite to them are also equal.
Given : ΔABC in which ∠B = ∠C
Prove that : AB = AC
Construction : Draw the angle bisector of ∠A and let it meet at D.
Statements 
Reasons 
1) ∠B = ∠C 
1) Given 
2) ∠BAD = ∠CAD 
2) By construction 
3) AD = AD 
3) Reflexive (common side) 
4) ΔABD ≅ ΔACD 
4) By angle angle side postulate (AAS) 
5) AB = AC 
5) By CPCTC 
Examples :
1) If ΔABC is an isosceles triangle with AB = AC. Prove that the perpendiculars from the vertices B and C to their opposite sides are equal.
Given : ΔABC is an isosceles triangle with AB = AC.
Prove that : BD = CE
Statements 
Reasons 
1) AB = AC 
1) Given 
2) ∠ABC = ∠ACB 
2) If two sides are congruent then the angle opposite to them are also congruent 
3) ∠CEB = ∠BDC 
3) Each 90^{0}

4) BC = BC 
4) Reflexive (common side) 
5) ΔBCE ≅BCD 
5) By AAS postulate 
6) BD = CE 
6) CPCTC 
2) ∠A = ∠C and AB = BC. Prove that ΔABD ≅ ΔCBE.
Given : ∠A = ∠C and AB = BC
Prove that : ΔABD ≅ ΔCBE
Statements 
Reasons 
1) ∠A = ∠C 
1) Given 
2) ∠AOE = ∠COD 
2) Vertically opposite angles 
3) ∠A + ∠AOE = ∠C + ∠COD 
3) Add (1) and (2) 
4) 180^{0}  ∠AEO = 180^{0}  ∠CDO

4) Since ∠A + ∠AOE + ∠AEO = 180 and ∠C + ∠COD + ∠CDO = 180 
5) ∠AEO = ∠CDO 
5) By subtraction property 
6) ∠AEO + ∠OEB = 180^{0}

6) Linear pair angles 
7) ∠CDO + ∠ODB = 180^{0}

7) Linear pair angles 
8) ∠AEO + ∠OEB = ∠CDO + ∠ODB 
8) Transitive property 
9) ∠OEB = ∠ODB 
9) Subtraction property and from(5) 
10) ∠CEB = ∠ADB 
10) Since ∠OEB = ∠CEB and ∠ODB = ∠ADB 
11) AB = BC 
11) Given 
11) ΔABD ≅ ΔCBE 
11) By AAS postulate (from (1),(10)) 
•
Side Angle Side Postulate
• Side Side Side Postulate
• Angle Angle Side Postulate
• Angle Side Angle Postulate
• HL postulate(Hypotenuse – Leg OR RHS)
From Angle Angle Side to Postulates of Congruent Triangle
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