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In the figure given below, the sides AB and AC of ∆ABC are extended to points P and Q respectively. If $∠ P B C < ∠ Q C B$ , then which of the following is true?

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A C = A B

A C < A B

A C > A B

None of these

answer is C.

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Detailed Solution

Given that sides AB and AC of

Δ A B C

are extended to points P and Q respectively. Also

∠ P B C < ∠ Q C B .

Here, AP and AQ are straight line and we know that the value of angle of a straight line is

180 ° .

∴ ∠ A B P = 180 °

and

∠ A C Q = 180 °

⇒ ∠ A B P = ∠ A C Q

⇒ ∠ A B C + ∠ C B P = ∠ A C B + ∠ B C Q from figure

⇒ ∠ C B P − ∠ B C Q = ∠ A C B − ∠ A B C (1)

Also,

∠ P B C < ∠ Q C B

⇒ ∠ P B C − ∠ Q C B < 0 (2)

Putting the value of (2) in (1), we get,

⇒ ∠ A C B − ∠ A B C = ∠ C B P − ∠ B C Q

⇒ ∠ A C B − ∠ A B C < 0

⇒ ∠ A C B < ∠ A B C (3)

We know that the side opposite to the greater angle is large.

⇒ A C > A B

Thus, if

∠ P B C < ∠ Q C B,

then

A C > A B .

Therefore, option (3) is true.

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