Angle Angle Side Postulate
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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.
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| 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
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| 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
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| 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)
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