Exterior Angle Bisector Theorem
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Exterior angle bisector theorem : The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle.Given : A ΔABC, in which AD is the bisector of the exterior ∠A and intersects BC produced in D.
Prove that : BD / CD = AB / AC
Construction : Draw CE || DA meeting AB in E.
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| 1) CE || DA | 1) By construction |
| 2) ∠1 = ∠3 | 2) Alternate interior angle |
| 3) ∠2 = ∠4 | 3) Corresponding angle (CE ||DA and BK is a transversal |
| 4) AD is a bisector of ∠A | 4) Given |
| 5) ∠1 = ∠2 | 5) Definition of angle bisector |
| 6) ∠3 = ∠4 | 6) Transitivity (from 2 and 4) |
| 7) AE = AC | 7) If angles are equal then side opposite to them are also equal |
| 8) BD / CD = BA/EA | 8) By Basic proportionality theorem(EC ||AD) |
| 9) BD /CD = AB/AE | 9) BA = AB and EA = AE |
| 10) BD /CD = AB /AC | 10) AE = EC and from(7) |
Examples
1) In the given figure, AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.
Given : AB = 10 cm, AC = 6 cm and BC = 12 cm
By exterior angle bisector theorem
BE / CE = AB / AC
(12 + x) / x = 10 / 6
6( 12 + x ) = 10 x [ by cross multiplication]
72 + 6x = 10x
<
72 = 10x – 6x
72 = 4x
x = 72/4
x = 18
CE = 18 cm
2) The bisector of interior ∠A of ΔABC meets BC in D, and the bisector of exterior ∠A meets BC produced in E. Prove that BD / BE = CD / CE.
Given : In ΔABC, AD and AE are respectively the bisectors of the interior and exterior angles at A.
Prove that : BD/BE = CD/CE
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| 1) AD is the internal bisector of ∠A | 1) Given |
| 2) AB / AC = BD / DC | 2) By Internal angle bisector theorem |
| 3) AE is an external bisector of ∠A | 3) Given |
| 4) AB/AC = BE / CE | 4) By External angle bisector theorem |
| 5) BD / DC = BE / CE | 5) From (2) and (4) |
| 6) BD/BE = CD/CE | 6) By alternendo ( a/b=c/d ⇒ a/c = b/d) |
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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