Chapter 6, Problem 14RP

To determine

To prove: The line segment RT is congruent to RS.

Explanation of Solution

Given information

A tringle ABC lies in given plane n

PA is congruent to PC.

AB is congruent to BC.

T and S are midpoints.

Proof:

  $\begin{array}{l}\overline{\text{PA}}\cong \overline{\text{PC}\text{.}}\mathrm{.......}\text{given}\\ \overline{\text{PA}}\cong \overline{\text{PB}}\mathrm{.........}\text{Reflexive property}\mathrm{......}\left(i\right)\\ \overline{\text{PT}}\cong \overline{\text{TA}}\mathrm{......}\text{given}\\ \overline{\text{PS}}\cong \overline{\text{SC}}\mathrm{.......}\text{given}\\ \overline{\text{RP}}=\overline{\text{PS}}\mathrm{.......}\text{Reflexive property}\mathrm{.......}\left(ii\right)\\ \overline{\text{AB}}\cong \overline{\text{BC}}\mathrm{......}\text{given}\mathrm{........}\left(iii\right)\\ \end{array}$  $\overline{\text{RT}}\cong \overline{\text{RS}}$

Thus, $\Delta \text{PRT }\cong \Delta \text{PST }$

…..SSS criteria is satisfied

Hence, $\overline{\text{RT}}\cong \overline{\text{RS}}$ …..Congruent part of congruent triangles are congruent