Question

given: ∠tuw ≅ ∠srw; \overline{rs} ≅ \overline{tu}
prove: ∠rst ≅ ∠uts
complete the paragraph proof:
it is given that ∠tuw ≅ ∠srw and \overline{rs} ≅ \overline{tu}.
because ∠rws and ∠uwt are vertical angles and
vertical angles are congruent, ∠rws ≅ ∠uwt. then,
by aas, △tuw ≅ △srw. because cpctc, \overline{sw} ≅
\overline{tw} and \overline{wu} ≅ \overline{rw}. because of the definition of
congruence, sw = tw and wu = rw. if we add those
equations together, sw + 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} ≅ \overline{tr}. because of
\boxed{} △trs ≅ △sut, and because of
\boxed{}, ∠rst ≅ ∠uts.

Explanation:

Step1: Identify First Congruence Criterion

We know \( \overline{RS} \cong \overline{TU} \) (given), \( \overline{SU} \cong \overline{TR} \) (proven), and we need to check the included angle or another side/angle. Wait, actually, we have \( \overline{RS} \cong \overline{TU} \), \( \overline{TR} \cong \overline{SU} \), and \( \overline{TS} \) is common? Wait, no, let's see: \( \triangle TRS \) and \( \triangle SUT \): sides \( TR \cong SU \), \( RS \cong TU \), and \( TS \cong ST \) (reflexive property). So by SSS (Side - Side - Side) congruence criterion, \( \triangle TRS \cong \triangle SUT \).

Step2: Identify CPCTC for Angles

Once we have \( \triangle TRS \cong \triangle SUT \), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the corresponding angles \( \angle RST \) and \( \angle UTS \) are congruent.

Answer:

First box: SSS (or Side - Side - Side)
Second box: CPCTC (or Corresponding Parts of Congruent Triangles are Congruent)