Question

proof:
\\(\overline{rs} \cong \overline{ut}\\)
\\(st \cong ts\\)
given
\\(\triangle rst \cong \triangle uts\\)
reflexive prop.
transitive prop
given
sss
sas
(diagram of quadrilateral rstu with triangle rst and uts, markings for congruent sides)

Explanation:

Step1: Analyze Given Information

We know \( \overline{RS} \cong \overline{UT} \) (given), \( \overline{ST} \cong \overline{TS} \) (reflexive property of congruence, since a segment is congruent to itself), and we need the third pair of sides for SSS congruence. From the diagram, \( \overline{RT} \cong \overline{US} \) (since the sides \( RT \) and \( US \) are marked with two ticks, indicating congruence).

Step2: Identify Congruence Criterion

To prove \( \triangle RST \cong \triangle UTS \) by SSS (Side - Side - Side) congruence criterion, we need three pairs of congruent sides. We already have \( \overline{RS} \cong \overline{UT} \), \( \overline{ST} \cong \overline{TS} \), and now we identify that \( \overline{RT} \cong \overline{US} \) (given by the markings in the diagram). So the missing given should be \( \overline{RT} \cong \overline{US} \), and the congruence criterion used is SSS (since we have three pairs of congruent sides: \( \overline{RS} \cong \overline{UT} \), \( \overline{ST} \cong \overline{TS} \), \( \overline{RT} \cong \overline{US} \)).

Answer:

The missing given is \( \boldsymbol{\overline{RT} \cong \overline{US}} \) and the congruence criterion for \( \triangle RST \cong \triangle UTS \) is SSS (Side - Side - Side) congruence criterion. The reflexive property is used for \( \overline{ST} \cong \overline{TS} \) (since \( \overline{ST} \) and \( \overline{TS} \) are the same segment, so they are congruent by reflexive property).