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Tell whether the following statement is true or false:

In a triangle ABC, the internal bisectors of ∠B and ∠C meet at P and the external bisectors of ∠B and ∠C meet at Q. Then ∠BPC + ∠BQC = 180°.

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True

False

answer is A.

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Detailed Solution

Given a triangle ABC in which the internal bisectors of ∠B and ∠C meet at P and the external bisectors of ∠B and ∠C meet at Q.
Question Image

As we know that, bisectors of

∠ B a n d ∠ C

meet at P.
Therefore,

∠ B P C = 90 ° + ∠ A 2 … … (1)

Exterior angle property states that if a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Also, using exterior angle property,

∠ B Q C = 90 ° − ∠ A 2 … … (2)

Adding both equations,

⇒ ∠ B P C + ∠ B Q C = 90 ° + ∠ A 2 + 90 ° − ∠ A 2

⇒ ∠ B P C + ∠ B Q C = 90 ° + 90 °

⇒ ∠ B P C + ∠ B Q C = 180 °

Given statement is true.
Hence, option 1 is correct.

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