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Bisectors of interior angle B nd exterior angle ACD of a ∆ ABC intersect at the point T. Prove that angle BTC=1/2 angle BAC.?
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Bisectors of interior angle B nd exterior angle ACD of a ∆ ABC interse...
Proof: Angle BTC = 1/2 Angle BAC


Given

ABC is a triangle with bisectors of interior angle B and exterior angle ACD intersecting at point T.


To Prove

Angle BTC = 1/2 Angle BAC


Proof

Let us consider the triangle ABC:


  • Let angle BAC be denoted by x.

  • Let angle ABC be denoted by y.

  • Let angle BCA be denoted by z.



By the angle bisector theorem,


  • Angle ABT = Angle TBC = y/2 (as BT is the angle bisector of angle B).

  • Angle ATC = (180 - z)/2 = 90 - z/2 (as TC is the angle bisector of angle B).



Also, angle ACD = x + y (as angle ACD is the exterior angle of angle B).


Hence, angle BTC = Angle ABT + Angle ATC = y/2 + 90 - z/2.


Now, let us look at triangle ACD:


  • Angle ADC = x + y (as it is the exterior angle of angle B).

  • Angle DAC = 180 - z (as the angles of triangle ADC add up to 180).



By the exterior angle theorem, angle ACT = Angle ADC + Angle DAC = x + y + 180 - z.


But, we know that angle ATC = 90 - z/2. Hence, angle ACT = 2(90 - z/2) + x + y - 180 = x + y - z.


Therefore, angle BTC = y/2 + 90 - z/2 = (y + x)/2 = 1/2 Angle BAC.


Conclusion

Thus, we have proved that angle BTC = 1/2 Angle BAC.
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Bisectors of interior angle B nd exterior angle ACD of a ∆ ABC intersect at the point T. Prove that angle BTC=1/2 angle BAC.?
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Bisectors of interior angle B nd exterior angle ACD of a ∆ ABC intersect at the point T. Prove that angle BTC=1/2 angle BAC.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Bisectors of interior angle B nd exterior angle ACD of a ∆ ABC intersect at the point T. Prove that angle BTC=1/2 angle BAC.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Bisectors of interior angle B nd exterior angle ACD of a ∆ ABC intersect at the point T. Prove that angle BTC=1/2 angle BAC.?.
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