Question No 45.In ΔPQR, points S and T lie on side PQ, and point U lies on side PR such that SP = SU = ST, QT = TU, and QU = QR. If ∠PQR = 2∠PRQ, then find ∠QUS.1)  36°

Given that ΔPQR is a triangle with points S and T on side PQ, and point U on side PR. Also, SP = SU = ST, QT = TU, and QU = QR.

Step 1: Since SP = SU, we can say that ∠PSU = ∠SUT (angles opposite to equal sides are equal). Let's denote this angle as 'x'.

Step 2: Similarly, since QT = TU, we can say that ∠QTU = ∠TUS. Let's denote this angle as 'y'.

Step 3: Now, in ΔPQR, we know that ∠PQR = 2∠PRQ. This implies that ∠PQR = 180° - 2∠PRQ (since the sum of angles in a triangle is 180°).

Step 4: Also, ∠PQR = ∠PQS + ∠SQT + ∠TQR (since S and T are points on PQ and QR respectively).

Step 5: Substituting the values from steps 1 and 2, we get ∠PQR = x + y + x.

Step 6: Substituting the value from step 3, we get 180° - 2∠PRQ = 2x + y.

Step 7: Now, in ΔPQR, ∠PRQ = ∠PSU + ∠SUT + ∠TUR (since S, U and T are points on PR, SU and TR respectively).

Step 8: Substituting the values from steps 1 and 2, we get ∠PRQ = x + y + y.

Step 9: Substituting the value from step 7 in step 6, we get 180° - 2(x + 2y) = 2x + y.

Step 10: Solving this equation, we get y = 36°.

Therefore, ∠QUS = y = 36°.