Question

ΔPQR is given and the sides QP and RP have been produced to S and T such that PQ = PS and PR = PT.Prove that the segment QR || ST​

Answer 1

Answer:

given and the sides QP and RP have been produced to S and T such that PQ = PS

Answer 2

$\displaystyle{\underline{\bf{Given:-}}}$

➔PQ = PS

➔PR = PT

$\displaystyle{\underline{\bf{Prove\:That:-}}}$

➔QR || ST

$\displaystyle{\underline{\bf{Diagram:-}}}$

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$\displaystyle{\underline{\bf{Solution:-}}}$

In ∆PQR and ∆PST

PQ = PS [Given]

PR = PT [Given]

$\angle$ QPR = $\angle$ SPT [Vertical Opposite Angle]

So, By SAS ∆PQR $\cong$ ∆PST

Now, by CPCT (Corresponding Parts of Congruent Triangle)

$\angle$ PQR = $\angle$ PST

These, are also pairs of Alternate Angles and we, know that Alternate Angles are made only between two parallel lines.

So, QR || ST [By Alternate Interior Angle]

QR || ST $\pink\bigstar$

Hence, Proved