Angle Angle Side Postulate

Angle angle side postulate (AAS)-> If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included 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 900
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) 1800 - ∠AEO = 1800 - ∠CDO 4) Since ∠A + ∠AOE + ∠AEO = 180
and ∠C + ∠COD + ∠CDO = 180
5) ∠AEO = ∠CDO 5) By subtraction property
6) ∠AEO + ∠OEB = 1800 6) Linear pair angles
7) ∠CDO + ∠ODB = 1800 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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