Question

given ∠tuw ≅ ∠srw, rs ≅ tu
prove ∠rst ≅ ∠uts
complete the paragraph proof
it is given that ∠tuw ≅ ∠srw and rs ≅ tu
because ∠rws and ∠uwt are vertical angles and
vertical angles are congruent, ∠rws ≅ ∠uwt then
by aas, △tuw ≅ △srw because cpctc, sw ≅
tw and wu ≅ 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 su ≅ tr because
of △trs ≅ △sut, and because of
∠rst ≅ ∠uts

Explanation:

Step1: Analyze the first blank

We have \( RS\cong TU\), \(SU\cong TR\) (proven \(SU = TR\) so \(SU\cong TR\)) and \(ST\) is a common side. For \(\triangle TRS\) and \(\triangle SUT\), by the \(SSS\) (Side - Side - Side) congruence criterion (\(RS = TU\), \(SU=TR\), \(ST = ST\)), \(\triangle TRS\cong\triangle SUT\).

Step2: Analyze the second blank

Since \(\triangle TRS\cong\triangle SUT\), and we want to show \(\angle RST\cong\angle UTS\). By the \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent) property, if two triangles are congruent, then their corresponding angles are congruent. So \(\angle RST\cong\angle UTS\) because they are corresponding parts of congruent triangles \(\triangle TRS\) and \(\triangle SUT\).

Answer:

First blank: \( SSS \) (Side - Side - Side congruence criterion). Second blank: \( CPCTC \) (Corresponding Parts of Congruent Triangles are Congruent).