Скачать или смотреть Class 10 | Property of an angle bisector of a triangle | Angle Bisector Theorem | Detail Explanation

In this video we will study proof of Class 10 | Property of an angle bisector of a triangle | Angle Bisector Theorem | Detail Explanation
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Property of an angle bisector of a triangle
Theorem : The bisector of an angle of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
Given: In Δ ABC, bisector of ∠C intersects seg AB in the point E
To prove: 𝐴𝐸/𝐸𝐵=𝐶𝐴/𝐶𝐵
Construction: Draw a line parallel to ray CE, passing through the point B. Extend AC so as to intersect it at point D.
Proof: ray CE ∥ ray BD and AD is transversal,
∴ ∠ACE ≅ ∠CDB ...(i)
Now taking BC as transversal
∠ECB ≅ ∠CBD ...(ii)
But, ∠ACE ≅ ∠ECB ...(iii)
∴ ∠CBD ≅ ∠CDB ... [from (I), (II) and (III)]
In ∆CBD, side CB ≅ side CD
∴ CB = CD ...(iv)
Now in Δ ABD, seg EC ∥ seg BD
∴ 𝐴𝐸/𝐸𝐵=𝐴𝐶/𝐶𝐷 ...(v)
∴ 𝐴𝐸/𝐸𝐵=𝐶𝐴/𝐶𝐵 ... [from (IV) and (V)]
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