Question

In the given figure, if TV = VU, QU = RT and ∠QTS = ∠PUR, then which of the following is not correct?

• A. ΔPQU ≅ ΔSRT
• B. ΔSVR ≅ ΔPVQ
• C. PQ + QR + PS + RS = 2(PV + QV + RS)
• D. ΔTQP ≅ ΔUSR

Answer 1

The correct option is D ΔTQP ≅ ΔUSR

Joining QS.
Let ∠QTS = ∠PUR = x.
Now, ∠STV + ∠QTS = 180° (Linear pair)
⇒ ∠STV = 180° – x
And, ∠PUV + ∠PUR = 180° (Linear pair)
⇒ ∠PUV = 180° – x
∴ ∠PUV = ∠STV …..(i)
Since they form a pair of alternative interior angles.
⇒ PU || TS
⇒ ∠UPV = ∠TSV (Alternative interior angles) …..(ii)
In ΔPVU and ΔSVT,
∠PUV = ∠STV [From (i)]
VU = VT (Given)
∠UPV = ∠TSV [From (ii)]
∴ ΔPVU ≅ ΔSVT (ASA congruence rule)
⇒ TS = PU (CPCT) …..(iii)
⇒ PV = VS (CPCT) …..(iv)
In ΔPQU and ΔSRT,
QU = RT (Given)
∠PUQ = ∠STR [From (i)]
PU = ST [From (iii)]
∴ ΔPQU ≅ ΔSRT (SAS congruence rule)
⇒ PQ = SR (CPCT) …..(v)
Now, in ΔSVR and ΔPVQ,
VR = VQ (∴ TV = VU and QU = RT)
∠RVS = ∠QVP (Vertically opposite angles)
VS = VP [From (iv)]
∴ ΔSVR ≅ ΔPVQ (SAS congruence rule)
Now, VR = VQ,
∴ QR = 2VQ …..(vi)
Also, PV = VS,
∴ PS = 2PV …..(vii)
From (v), (vi) and (vii), we get
PQ + QR + PS + RS = RS + 2QV + 2PV + RS
= 2(QV + PV + RS)
Thus, only ΔTQP ≅ ΔUSR is incorrect.
Hence, the correct answer is option (d).