Question

Given: RT ≅ TV and ST ≅ TU
Prove: RSVU is a parallelogram.

Quadrilateral R S V U is shown. Diagonals are drawn from point R to point V and from point S to point U and intersect at point T. The lengths of R T and T V are congruent. The lengths of U T and T S are congruent.


Statements

Reasons
1. RT ≅ TV;
ST ≅ TU 1. given
2. ∠RTS and ∠VTU are vert. ∠s;
∠RTU and ∠VTS are vert. ∠s 2. definition of vertical angles
3. ∠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

Answer 1

△RTS ≅ △VTU and △RTU ≅ △VTS

(Using RT ≅ TV, ST ≅ TU and the pairs of vertical angles ∠RTS ≅ ∠VTU and ∠RTU ≅ ∠VTS, each pair of triangles is congruent by SAS.)