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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.

Bisectors of the Angles of a Triangle Meet at a Point

Proof:

          Statement

1. In ∆XOC and ∆XOB,

(i) ∠CXO = ∠BXO

(ii) ∠XCO = XBO = 90°

(iii) XO = XO.

 

2. ∆XOC ≅ ∆XOB

3. OC = OB

4. Similarly, ∆YOC ≅ ∆YOA

5. OC = OA

6. OB = OA.

7. In ∆ZOA and ∆ZOB,

(i) OA = OB

(ii) OZ = OZ

(iii) ∠ZAO = ∠ZBO = 90


8. ∆ZOA ≅ ∆ZOB.

9. ∠ZOA = ∠ZOB.

10. NO bisects ∠XZY. (Proved)

          Reason

1.

(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.





9th Grade Math

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