# Bisectors of the Angles of a Triangle Meet at a Point

Here we will prove that the bisectors of the angles of a triangle meet at a point.

Solution:

Given In ∆XYZ, XO and YO bisect ∠YXZ and ∠XYZ respectively.

To prove: OZ bisects ∠XZY.

Construction: Draw OA ⊥ YZ, OB ⊥ XZ and OC ⊥ XY.

Proof:

 Statement1. In ∆XOC and ∆XOB,(i) ∠CXO = ∠BXO(ii) ∠XCO = XBO = 90°(iii) XO = XO. 2. ∆XOC ≅ ∆XOB3. OC = OB4. Similarly, ∆YOC ≅ ∆YOA5. OC = OA6. OB = OA.7. In ∆ZOA and ∆ZOB,(i) OA = OB(ii) OZ = OZ(iii) ∠ZAO = ∠ZBO = 908. ∆ZOA ≅ ∆ZOB.9. ∠ZOA = ∠ZOB. 10. NO bisects ∠XZY. (Proved) Reason1.(i) XO bisects ∠YXZ(ii) Construction.(iii) Common Side. 2. By AAS criterion of congruency.3. CPCTC.4. Proceeding as above.5. CPCTC.6. Using statement 3 and 5.7.(i) From Statement 6.(ii) Common Side.(iii) Construction. 8. By RHS criterion of congruency.9. CPCTC. 10. From statement 9.

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