Interior Angle Bisector Theorem

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)

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Examples

1) In the given figure, AD is the bisector of ∠A. If BD = 4 cm,
DC = 3 cm and AB = 6 cm, find AC.

Solution :
In Δ ABC, AD is the bisector of ∠A.

∴ BD / DC = AB / AC ( Angle bisector theorem)

⇒ 4 / 3 = 6 / AC

⇒ 4 AC = 18

⇒ AC = 18 / 4 = 4.5 cm.

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2) AD is a
median of ΔABC. The bisector of ∠ADB and ∠ADC meet AB and AC in E and F respectively.

Given : In ΔABC, AD is the median and DE and DF are angle bisectors of ∠ADB and ∠ADC respectively.

Prove that : EF || BC

Statements
Reasons
1) DE is the angle bisector of ∠ADB 1) Given
2) ∴ AD / DB = AE / EB 2) By interior angle bisector theorem
3) DF is the angle bisector of ∠ADC 3) Given
4) AD / DC = AF / FC 4) By angle bisector theorem
5) AD is the median 5) Given
6) BD = DC 6) By definition of median
7) AD / DB = AF / FC 7) From (6)
8) AE / EB = AF / FC 8) From (2) and (7)
Hence EF || BC

Similarity in Triangles

Similarity in Geometry
Properties of similar triangles
Basic Proportionality Theorem(Thales theorem)
Converse of Basic Proportionality Theorem
Interior Angle Bisector Theorem
Exterior Angle Bisector Theorem
Proofs on Basic Proportionality
Criteria of Similarity of Triangles
Geometric Mean of Similar Triangles
Areas of Two Similar Triangles

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