Step 1

It is given that,

\(\displaystyle\angle{Q}\stackrel{\sim}{=}\angle{S}\)

\(\displaystyle\overline{{{Q}{R}}}\stackrel{\sim}{=}\overline{{{S}{R}}}\)

\(\displaystyle{I}{n}\triangle{P}{Q}{R}{\quad\text{and}\quad}\triangle{R}{S}{T}\),

\(\displaystyle\angle{Q}\stackrel{\sim}{=}\angle{S}\) (equal angles given)

\(\displaystyle\overline{{{Q}{R}}}\stackrel{\sim}{=}\overline{{{S}{R}}}\) ( equal sides given)

\(\displaystyle\angle{P}{R}{Q}\stackrel{\sim}{=}\angle{S}{R}{T}\) ( vertically opposite angles)

Here, two angles and one side of both triangle is equal then,

Step 2

\(\displaystyle\triangle{P}{Q}{R}\stackrel{\sim}{=}\triangle{R}{S}{T}\) (By angle-side-angle congruence)

Hence option C is correct.

It is given that,

\(\displaystyle\angle{Q}\stackrel{\sim}{=}\angle{S}\)

\(\displaystyle\overline{{{Q}{R}}}\stackrel{\sim}{=}\overline{{{S}{R}}}\)

\(\displaystyle{I}{n}\triangle{P}{Q}{R}{\quad\text{and}\quad}\triangle{R}{S}{T}\),

\(\displaystyle\angle{Q}\stackrel{\sim}{=}\angle{S}\) (equal angles given)

\(\displaystyle\overline{{{Q}{R}}}\stackrel{\sim}{=}\overline{{{S}{R}}}\) ( equal sides given)

\(\displaystyle\angle{P}{R}{Q}\stackrel{\sim}{=}\angle{S}{R}{T}\) ( vertically opposite angles)

Here, two angles and one side of both triangle is equal then,

Step 2

\(\displaystyle\triangle{P}{Q}{R}\stackrel{\sim}{=}\triangle{R}{S}{T}\) (By angle-side-angle congruence)

Hence option C is correct.