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State whether the following statement is true or false. The external bisector of an angle of a triangle divides the opposite sides externally in the ratio of the sides containing the angle.
A. True
B. False

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Last updated date: 24th Jul 2024
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Answer
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Hint: To solve this question we need to have a basic knowledge about the properties of triangles and Thales theorem and a basic idea about the construction part which would help us to solve this problem in a simple way by using the required theorems.

Complete step-by-step solution:
According to the question it is asked of us to state whether the statement, ‘the external bisector of an angle of a triangle divides the opposite sides externally in the ratio of the sides containing the angle’ is true or false. If we look at this statement for external bisectors and externally in the ratio of the sides, then:
Let us consider a triangle ABC, in which AD is the bisector of the exterior $\angle A$ and intersects BC produced to D. Then we have to prove $\dfrac{BD}{CD}=\dfrac{AB}{AC}$ if we have to prove the statement as true.
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Now by the construction of this triangle ABC it is clear that $CE\parallel AD$ and we can say $\angle 1=\angle 3$ because these are alternate interior angles. And $\angle 2=\angle 4$ because $CE\parallel DA$ and BE is a transversal. And it is given to us that AD is a bisector of $\angle A$. Now by definition of angle bisector we can say that $\angle 1=\angle 2$ and transitivity (from 2 and 4) $\angle 3=\angle 4$. If the angles are equal then the side opposite to them are also equal, so,
$AE=AC\ldots \ldots \ldots \left( i \right)$
By basic proportionality theorem $\left( EC\parallel AD \right)$, we will write as,
$\dfrac{BD}{CD}=\dfrac{BA}{EA}$
And since, BA = AB and EA = AE, so,
$\dfrac{BD}{CD}=\dfrac{AB}{AE}$
And we have AE = AC from (i), then,
$\dfrac{BD}{CD}=\dfrac{AB}{AC}$
Hence it is proved and the correct option is A.

Note: While solving these type of questions you have to keep in mind all the general properties of a triangle because all those will help to solve these type of questions and all the basic theorems for angles and sides are also to be kept in mind exactly, otherwise it will be tough to solve the questions.