Question

internl bisector therom

Answer 1

Interior Angle Bisector Theorem : The angle bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Given : A ΔABC in which AD is the internal bisector of ∠A and meets BC in D.
Prove that : BD / DC = AB / AC 
Construction : Draw CE || DA to meet BA produced in E.




Statements

Reasons

1) CE || DA

1) By construction

2) ∠2 = ∠3

2) Alternate interior angles

3) ∠1 = ∠4

3) Corresponding angles

4) AD is the bisector

4) Given

5) ∠1 =∠2

5) Definition of angle bisector

6) ∠3= ∠4

6) From (2) and (3)

7) AE = AC

7) In ΔACE, side opposite to equal angles are equal

8) BD / DC = BA / AE

8) In ΔBCE DA || CE and by BPT theorem

9) BD / DC = AB / AC

9) From (7)