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Question

State whether the statements are True or False: The measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.___

Answer

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Solution

Let a \(\Delta ABC\) be a given triangle such that side \(BC\) is extended to \(D\), as shown in the figure below:
Solution for State whether the statements are True or False: The measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.___
Now, \(BCD\) is a straight line, so, \(\angle 2 \;\&\;\angle 1\) are linear pair.
\(\implies \angle 1+\angle 2=180^{\circ}...(1)\)
Now, we know that the sum of interior angles of a triangle is \(180^{\circ}\).
In \(\Delta ABC\), we have,
\(\implies \angle 2+\angle 3+\angle 4=180^{\circ}...(2)\)
Using \((1)\) and \((2)\),
\(\implies \angle 1+\angle 2=\angle 2+\angle 3+\angle 4\)
\(\implies \angle 1=\angle 3+\angle 4\)
Now, \(\angle 1\) is an exterior angle and \(\angle 3,\angle 4\)are interior angles of \(\Delta ABC\).
Therefore, it proves that the measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.
Hence, the given statement is true.
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