We are asked to prove that triangles △PQR and △TSR are congruent.
Let us write the statements and reasons to prove that the given triangles are congruent.
[tex]\begin{gathered} 1.\: Statement\colon\: \bar{PQ}\cong\bar{TS} \\ 1.Reason\colon Given \end{gathered}[/tex]
We are given that R is the midpoint of QS and PT
This means that QR ≅ SR and also PR ≅ TR by the property of "Definition of midpoint"
[tex]\begin{gathered} 2.\: Statement\colon\: \bar{QR}\cong\bar{SR} \\ 2.\: Reason\colon\: \text{Definition of midpoint} \end{gathered}[/tex][tex]\begin{gathered} 3.\: Statement\colon\: \bar{PR}\cong\bar{TR} \\ 3.\: Reason\colon\: \text{Definition of midpoint} \end{gathered}[/tex]
So, now we have 3 equal sides in both triangles
Therefore, by the property of "Side-Side-Side" (SSS) the given triangles are congruent.
[tex]\begin{gathered} 4.\: Statement\colon\: \triangle PQR\cong\triangle TSR \\ 4.\: Reason\colon\: \text{Side}-\text{Side}-\text{Side theorem of triangle congruence } \end{gathered}[/tex]
