# 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.

From Bisectors of the Angles of a Triangle Meet at a Point to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Addition and Subtraction of Fractions | Solved Examples | Worksheet

Jul 18, 24 03:08 PM

Addition and subtraction of fractions are discussed here with examples. To add or subtract two or more fractions, proceed as under: (i) Convert the mixed fractions (if any.) or natural numbers

2. ### Worksheet on Simplification | Simplify Expressions | BODMAS Questions

Jul 18, 24 01:19 AM

In worksheet on simplification, the questions are based in order to simplify expressions involving more than one bracket by using the steps of removal of brackets. This exercise sheet

3. ### Fractions in Descending Order |Arranging Fractions an Descending Order

Jul 18, 24 01:15 AM

We will discuss here how to arrange the fractions in descending order. Solved examples for arranging in descending order: 1. Arrange the following fractions 5/6, 7/10, 11/20 in descending order. First…

4. ### Fractions in Ascending Order | Arranging Fractions | Worksheet |Answer

Jul 18, 24 01:02 AM

We will discuss here how to arrange the fractions in ascending order. Solved examples for arranging in ascending order: 1. Arrange the following fractions 5/6, 8/9, 2/3 in ascending order. First we fi…