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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