Question

given: \\( \angle t u w \cong \angle s r w ; \overline{r s} \cong \overline{t u} \\) prove: \\( \angle r s t \cong \angle u t s \\) complete the paragraph proof: it is given that \\( \angle t u w \cong \angle s r w \\) and \\( \overline{r s} \cong \overline{t u} \\). because \\( \angle r w s \\) and \\( \angle u w t \\) are vertical angles and vertical angles are congruent, \\( \angle r w s \cong \angle u w t \\). then, by aas, \\( \triangle t u w \cong \triangle s r w \\). because cpctc, \\( \overline{s w} \cong \overline{t w} \\) and \\( \overline{w u} \cong \overline{r w} \\). because of the definition of congruence, \\( s w=t w \\) and \\( w u=r w \\). if we add those equations together, \\( s w+w u=t w+r w \\). because of segment addition, \\( s w+w u=s u \\) and \\( t w+r w=t r \\). then by substitution, \\( s u=t r \\). if segments are equal, then they are congruent, so \\( \overline{s u} \cong \overline{t r} \\). because of \\( \triangle t r s \cong \triangle s u t \\), and because of \\( \angle r s t \cong \angle u t s \\).

Explanation:

We have \( \overline{RS}\cong\overline{TU}\) (given), \( \overline{SU}\cong\overline{TR}\) (proven above), and \( \overline{ST}\cong\overline{TS}\) (common side). The Side - Angle - Side (SAS) congruence postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, in \( \triangle TRS\) and \( \triangle SUT\), the two sides \( \overline{RS}\cong\overline{TU}\), \( \overline{SU}\cong\overline{TR}\) and the included side \( \overline{ST}\cong\overline{TS}\) satisfy the SAS condition. Once \( \triangle TRS\cong\triangle SUT\) by SAS, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \( \angle RST\cong\angle UTS\).

Answer:

First blank: SAS, Second blank: CPCTC