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Question
Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.
Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.
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Solution
AL is bisector of angle A. Let D is any point on AL. From D, a straight line DE is drawn parallel to AC.
DE || AC [Given]
∠ADE = ∠DAC….(i) [Alternate angles]
∠DAC = ∠DAE…….(ii) [AL is bisector of ∠A]
From (i) and (ii)
∠ ADE = ∠ DAE
AE = ED [Sides opposite to equal angles are equal]
Therefore, AED is an isosceles triangle.
AL is bisector of angle A. Let D is any point on AL. From D, a straight line DE is drawn parallel to AC.
DE || AC [Given]
∠ADE = ∠DAC….(i) [Alternate angles]
∠DAC = ∠DAE…….(ii) [AL is bisector of ∠A]
From (i) and (ii)
∠ ADE = ∠ DAE
AE = ED [Sides opposite to equal angles are equal]
Therefore, AED is an isosceles triangle.
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