Question

proving the converse of the parallelogram diagonal theorem
given: ( overline{rw} cong overline{wt}; overline{uw} cong overline{ws} )
prove: rstu is a parallelogram.
identify the steps that complete the proof.

Explanation:

Step1: Reason for Statement 3

Vertical angles are congruent. So, the reason for $\angle SWR\cong\angle UWT$ (Statement 3) is "vertical angles theorem".

Step2: Reason for Statement 4

We have two sides and the included angle ( $\overline{RW}\cong\overline{WT}$, $\angle SWR\cong\angle UWT$, $\overline{UW}\cong\overline{WS}$ ) congruent. By the Side - Angle - Side (SAS) congruence criterion, $\triangle SWR\cong\triangle UWT$ (Statement 4). So the reason is "SAS congruence theorem".

Step3: Reason for Statement 5

Since $\triangle SWR\cong\triangle UWT$, corresponding parts of congruent triangles are congruent (CPCTC). So, $\angle WRS\cong\angle WTU$ and $\angle WSR\cong\angle WUT$ (Statement 5). The reason is "CPCTC".

Answer:

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