Question

the given figure, the side QR of ΔPQR is produced to a point S. If the bisector of ∠PQR and ∠PRS meet at point T, and ∠QPR=56° then the value of ∠QTR is
A) 27°
B) 122°
C) 28°
D) 90°​

Answer 1

Answer:

Option (c) is correct. 28°.

Step-by-step explanation:

To find: Value of ∠QTR .

Solution :

We have,

• ∠QPR is 56°.
• TR is bisecting ∠PRS. So, ∠PRT and ∠TRS are equal.
• TQ is bisecting ∠PQR. So, ∠PQT and ∠TQR are equal.

For finding value first we will prove that ∠QTR is half of ∠QPR. So,

We know,

✿ Sum of any two interior angles of triangle is equal to the opposite exterior angle.

So,

$\implies$ ∠TQR + ∠QTR = ∠TRS ...(i)

And,

$\implies$ ∠PQR + ∠QPR = ∠SRP ...(ii)

Take equation (i) :

$\implies$ ∠TQR + ∠QTR = ∠TRS

• We want value of ∠QTR. So,

$\implies$ ∠QTR = ∠TRS - ∠TQR ...(iii)

Then,

Take equation (ii) :

$\implies$ ∠PQR + ∠QPR = ∠SRP ...(ii)

• ∠PQR = ∠PQT + ∠TQR, Or, ∠PQR = 2∠TQR beacuse ∠PQT and ∠TQR are equal.
• Similarly, ∠SRP = ∠PRT + ∠TRS or ∠SRP = 2∠TRS beacuse ∠PRT and ∠TRS are equal.

$\implies$ 2∠TQR + ∠QPR = 2∠TRS

$\implies$ ∠QPR = 2∠TRS - 2∠TQR

$\implies$ ∠QPR = 2(∠TRS - ∠TQR) [From equation (iii) ∠TRS - ∠TQR = ∠QTR]

$\implies$ ∠QPR = 2∠QTR

$\implies$ ∠QTR = ∠QPR/2

• Put value of ∠QPR = 56° for finding value of ∠QTR.

$\implies$ ∠QTR = 56°/2

$\implies$ ∠QTR = 28°

Therefore,

Value of ∠QTR is 28°.