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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