Question

the proof that \\( \triangle rst \cong \triangle vst \\) is shown
given \\( \overline{st} \\) is the perpendicular bisector of \\( \overline{rv} \\)
prove \\( \triangle rst \cong \triangle vst \\)
what is the missing reason in the proof?
\\( \

$$\begin{array} { | l | l | } \\hline \\text { statements } & \\text { reasons } \\\\ \\hline 1 \\overline { s t } \\text { is the perpendicular } & 1 \\text { given } \\\\ \\text { bisector of } \\overline { r v } & \\\\ \\hline 2 \\angle s t r \\text { and } \\angle s t v \\text { are right } & 2 \\text { def of perpendicular } \\\\ \\text { angles } & \\text { bisector } \\\\ \\hline 3 \\overline { r s } \\cong \\overline { v s } & 3 ? \\\\ \\hline 4 \\overline { s t } \\cong \\overline { s t } & 4 \\text { reflexive property } \\\\ \\hline 5 \\triangle r s t \\cong \\triangle v s t & 5 \\text { hl theorem } \\\\ \\hline \\end{array}$$

\\)
perpendicular bisector theorem
converse of the perpendicular bisector theorem
pythagorean theorem
sss congruence theorem

Explanation:

Step1: Recall the perpendicular bisector theorem

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Step2: Apply the theorem to the problem

Since \(S\) is on the perpendicular bisector \(\overline{ST}\) of \(\overline{RV}\), by the perpendicular bisector theorem, \(RS = VS\) (i.e., \(\overline{RS}\cong\overline{VS}\)). So the reason for statement 3 is the perpendicular bisector theorem.

Answer:

perpendicular bisector theorem