state and prove angle bisector property of a triangle.
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Answers
Answer:
What is Angle Bisector Theorem?
An angle bisector is a straight line drawn from the vertex of a triangle to its opposite side in such a way, that it divides the angle into two equal or congruent angles.
Step-by-step explanation:
Proof
Given : In ΔABC, AD is the external bisector of ∠BAC and intersects BC produced at D.
To prove : BD/DC = AB/AC
Constt: Draw CE ∥ DA meeting AB at E
Exterior angle bisector theorem
Since, CE ∥ DA and AC is a transversal, therefore,
∠ECA = ∠CAD (alternate angles) ……(1)
Again, CE ∥ DA and BP is a transversal, therefore,
∠CEA = ∠DAP (corresponding angles) —–(2)
But AD is the bisector of ∠CAP,
∠CAD = ∠DAP —–(3)
As we know, Sides opposite to equal angles are equal, therefore,
∠CEA = ∠ECA
In ΔBDA, EC ∥ AD.
BD/DC = BA/AE [By Thales Theorem]
AE = AC,
BD/DC = BA/AC
Hence, proved.
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