Question

given: \\( \angle tuw \cong \angle srw; \overline{rs} \cong \overline{tu} \\)
prove: \\( \angle rst \cong \angle uts \\)
complete the paragraph proof:
it is given that \\( \angle tuw \cong \angle srw \\) and \\( \overline{rs} \cong \overline{tu} \\).
because \\( \angle rws \\) and \\( \angle uwt \\) are vertical angles and
vertical angles are congruent, \\( \angle rws \cong \angle uwt \\).
then, by aas, \\( \triangle tuw \cong \triangle srw \\). because cpctc,
\\( \overline{sw} \cong \overline{tw} \\) and \\( \overline{wu} \cong \overline{rw} \\). because of the definition
of congruence, \\( sw = tw \\) and \\( wu = rw \\). if we add
those equations together, \\( sw + wu = tw + rw \\).
because of segment addition, \\( sw + wu = su \\) and
\\( tw + rw = tr \\). then by substitution, \\( su = tr \\). if
segments are equal, then they are congruent, so \\( \overline{su} \\)
\\( \cong \overline{tr} \\). because of \\( \square \\), \\( \triangle trs \cong \triangle sut \\), and
because of \\( \square \\), \\( \angle rst \cong \angle uts \\).

Explanation:

Step1: Determine the congruence criterion for \(\triangle TRS\) and \(\triangle SUT\)

We know that \(RS\cong TU\) (given), \(SU\cong TR\) (proven in the paragraph), and \(ST = ST\) (common side). By the Side - Side - Side (SSS) congruence criterion, \(\triangle TRS\cong\triangle SUT\).

Step2: Use the property of congruent triangles

Since \(\triangle TRS\cong\triangle SUT\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, \(\angle RST\cong\angle UTS\).

Answer:

The first blank is \(SSS\) (Side - Side - Side congruence criterion) and the second blank is \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent).