In Fig. 6.9, OD is the bisector of ∠AOC, OE is the bisector of ∠BOC and OD ⊥ OE. Show that the points A, O and B are collinear
Solution:
Given, OD is the bisector of ∠AOC
OE is the bisector of ∠BOC
OD ⊥ OE
We have to show that A, O and B are collinear.
Since OD is the bisector of ∠AOC
∠AOD = ∠DOC
∠AOC = ∠AOD + ∠DOC
∠AOC = ∠DOC + ∠DOC
∠AOC = 2∠DOC ------------ (1)
Since OE is the bisector of ∠BOC
∠BOE = ∠EOC
∠BOC = ∠BOE + ∠EOC
∠BOC = ∠EOC + ∠DOC
∠BOC = 2∠EOC ------------ (2)
Adding (1) and (2),
∠AOC + ∠BOC = 2∠DOC + 2∠EOC
= 2(∠DOC + ∠EOC)
= 2∠DOE
Since OD ⊥ OE, ∠DOE = 90°
∠AOC + ∠BOC = 2(90°)
= 180°
From the figure,
∠AOC + ∠BOC = ∠AOB
∠AOB = 180°
This implies that A, O and B form a linear pair of angles.
Therefore, the points A, O and B lie on a straight line and they are collinear.
✦ Try This: In the given figure, if AB ∥ PQ,PR ∥ BC and ∠QPR=120°, what is the measure of ∠ABC?
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 6
NCERT Exemplar Class 9 Maths Exercise 6.3 Problem 1
In Fig. 6.9, OD is the bisector of ∠AOC, OE is the bisector of ∠BOC and OD ⊥ OE. Show that the points A, O and B are collinear
Summary:
In Fig. 6.9, OD is the bisector of ∠AOC, OE is the bisector of ∠BOC and OD ⊥ OE. It is shown that the points A, O and B are collinear as they form a linear pair of angles. The points A, O and B lie on a straight line
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