Question

∠rtu and ∠vts are vert. ∠s

1. ∠rts ≅ ∠vtu,

∠rtu ≅ ∠vts 3 vertical angles are congruent
4 ? 4 sas congruency theorem
5 ∠vrs ≅ ∠rvu, ∠usr ≅ ∠suv,
∠vru ≅ ∠rvs, ∠rus ≅ ∠usv 5 cpctc
6 ∠vrs and ∠rvu, ∠usr and ∠suv, ∠vru and ∠rvs, ∠rus and ∠usv
are each a pair of alternate interior angles 6 definition of alternate interior
angles
7 rs || uv and ru || sv 7 converse of the parallelogram
diagonal theorem
8 rsvu is a parallelogram 8 definition of parallelogram
what is the missing statement in step 4?
○ △rts ≅ △vtu and △rtu ≅ △vts
○ △rts ≅ △rvs and △rtu ≅ △stv
○ △vrs ≅ △vru and △usr ≅ △usv
○ △vur ≅ △vus and △uvs ≅ △sru

Explanation:

Step1: Recall SAS congruency theorem

The SAS (Side - Angle - Side) congruency theorem 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.

Step2: Analyze the given information

We know from step 3 that \(\angle RTS\cong\angle VTU\) and \(\angle RTU\cong\angle VTS\). If we assume that the sides \(RT = VT\) and \(ST=UT\) (not shown in the given text but implied by the use of the SAS theorem in the proof structure), then the two triangles \(\triangle RTS\) and \(\triangle VTU\) (and \(\triangle RTU\) and \(\triangle VTS\)) would satisfy the SAS condition.

Answer:

\(\triangle RTS\cong\triangle VTU\) and \(\triangle RTU\cong\triangle VTS\)