Question

proving the converse of the parallelogram diagonal theorem
given: rw ≅ wt, uw ≅ ws
prove: rstu is a parallelogram.
identify the steps that complete the proof.
statements

1. rw ≅ wt, uw ≅ ws
2. ∠swr and ∠uwt are vertical angles
3. ∠swr ≅ ∠uwt
4. △swr ≅ △uwt
5. ∠wrs ≅ ∠wtu, ∠wsr ≅ ∠wut
6. ru || st, ut || rs
7. rstu is a parallelogram

reasons

1. given
2. def. of vertical angles

3.
4.
5.

1. converse of alt. interior angles theorem
2. def. of a parallelogram

Explanation:

Step1: Recall vertical - angle property

Vertical angles are congruent. So, for $\angle SWR$ and $\angle UWT$ which are vertical angles, $\angle SWR\cong\angle UWT$ because vertical angles are congruent.

Step2: Determine triangle - congruence criterion

We have $\overline{RW}\cong\overline{WT}$, $\angle SWR\cong\angle UWT$, and $\overline{UW}\cong\overline{WS}$. By the Side - Angle - Side (SAS) congruence criterion, $\triangle SWR\cong\triangle UWT$.

Step3: Use CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Since $\triangle SWR\cong\triangle UWT$, we have $\angle WRS\cong\angle WTU$ and $\angle WSR\cong\angle WUT$.

Answer:

1. vertical angles are congruent
2. SAS
3. CPCTC