Question

Draw angle ABC of any suitable measure.

(i) Draw BP, the bisector of angle ABC.

(ii) Draw BR, the bisector of angle PBC and draw BQ, the bisector of angle ABP.

(iii) Are the angles ABQ, QBP, PBR and RBC equal ?

(iv) Are the angles ABR and QBC equal ?

Answer 1

Steps of Construction :

1. Construct any angle ABC

2. With B as centre, draw an arc EF meeting BC at E and AB at F.

3. With E, F as centres draw two arc of equal radii meeting each other at the point P.

4. Join BP. Then BP is the bisector of $\angle$ABC

$\therefore \angle \text{ABP}=\angle \text{PBC}=\frac{1}{2}\angle \text{ABC}$

5. Similarly draw BR, the bisector of $\angle$PBC and draw BQ as the bisector of $\angle$ABP

[With the same method as in steps 2, 3]

6. Then $\angle$ABQ=$\angle$QBP=$\angle$PBR=$\angle$RBC

7. $\angle$ABR=$\frac{3}{4}$$\angle$ABC and $\angle$QBC=$\frac{3}{4}$$\angle$ABC

$\therefore$$\angle$ABR=$\angle$QBC.