Question

Assertion (A): If one angle of a triangle is equal to one angle of another triangle and bisectors of these angles divide the opposite sides in the same ratio, then the triangles are similar.
Reason (R): The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Which of the following is true?

• A. (A) and (R) are true and (R) is the correct explanation of (A)
• B. (A) and (R) are true but (R) is not the correct explanation of (A)
• C. (A) is true and (R) is false
• D. Both (A) and (R) are false

Answer 1

The correct option is A

(A) and (R) are true and (R) is the correct explanation of (A)

Given AD bisects $\angle BAC$ and PS bisects $\angle QPR$
such that,
$\frac{BD}{DC}=\frac{QS}{SR}$ ...(i)
and $\angle BAC=\angle QPR$
Now, if AD bisects $\angle BAC$
Then, $\frac{BD}{DC}=\frac{AB}{AC}$ ...(ii)
[since, the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]
and if PS bisects $\angle QRP$
$\Rightarrow \frac{QS}{SR}=\frac{QP}{PR}$ ...(iii)
[since, the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]
From Eqs. (i),(ii) and (iii), we get
$\frac{AB}{AC}=\frac{QP}{PR}$
Also, $\angle BAC=\angle QPR$
$\therefore \Delta ABC\sim \Delta PQR$ [by SAS similarity]