Question

proving the converse of the parallelogram diagonal theorem
statements reasons

1. \\( \overline { rw } \cong \overline { wt } , \overline { uw } \cong \overline { ws } \\) 1. given
2. \\( \angle swr \\) and \\( \angle uwt \\) are vertical angles 2. def. of vertical angles
3. \\( \angle swr \cong \angle uwt \\) 3. \\( \blacksquare \\)
4. \\( \triangle swr \cong \triangle uwt \\) 4. \\( \diamond \\)
5. \\( \angle wrs \cong \angle wtu \\), \\( \angle wsr \cong \angle wut \\) 5. \\( \blacksquare \\)
6. \\( \overline { ru } \parallel \overline { st } , \overline { ut } \parallel \overline { rs } \\) 6. converse of alt. interior angles theorem

identify the steps that complete the proof.

Explanation:

Step1: Vertical angles theorem

Vertical angles are congruent. So, for step 3, the reason is "vertical angles theorem".

Step2: SAS (Side - Angle - Side) congruence

We have \( \overline{RW}\cong\overline{WT}\), \( \angle SWR\cong\angle UWT\), \( \overline{UW}\cong\overline{WS}\). By the Side - Angle - Side (SAS) congruence criterion, \( \triangle SWR\cong\triangle UWT\). So, for step 4, the reason is "SAS".

Step3: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle SWR\cong\triangle UWT\), their corresponding parts are congruent. So, \( \angle WRS\cong\angle WTU\) and \( \angle WSR\cong\angle WUT\). For step 5, the reason is "CPCTC".

Answer:

1. vertical angles theorem; 4. SAS; 5. CPCTC