Question

in a figure PQRS is a quadrilateral and T and U are respectively points on PS and RS such that PQ=RQ,angle PQT=angle RQU and angle TQS=angle UQS.prove that QT=QU.please answer fast!attachments included.NO SPAMMERS ​

Answer 1

Answer:

QU = QT

Step-by-step explanation:

∠ PQS = ∠PQT + ∠TQS

∠RQS = ∠RQU + ∠UQS

given that ∠PQT = ∠RQU & ∠TQS = ∠UQS

=> ∠ PQS = ∠RQS

now in ΔRQS & Δ PQS

PQ = RQ given

∠ PQS = ∠RQS

QS is common

=> ΔRQS ≅ Δ PQS

=> ∠QRU = ∠QPT

now in Δ RQU  & Δ PQT

RQ = PQ

∠RQU = ∠PQT

∠QRU = ∠QPT

=> Δ RQU  ≅ Δ PQT

=> QU = QT

Answer 2

Answer:

Here is your answer !!!

Step-by-step explanation:

RU+∠UQS

In △PQS and △RQS,

PQ=QR (given)

∠PQS=∠RQS

[ ∵∠PQT=∠RQU &

∠TQS=∠UQS

∴∠PQT+∠TQS=∠RQU+∠UQS

∠PQS=∠RQS ]

QS=QS [common]

∴△PQT≅△RQS [By SAS]

∠PSQ=∠QSR [By CPCT] →(1)

Now, in △QTS △QSU

QS=QS (common)

∠TQS=∠SQU [given]

∠TSQ=∠USQ [from (1)]

∴△QTS≅△QSU

∴QT=QU [By CPCT]