write a two column proof.

Given: RS=~ TS, V is the midpoint of RT
Prove: /\RSV=~ /\TSV

/\ is the triangle sign
=~ is the congruent sign

you're given that RS≅TS

SV equals itself (reflection)

RV equals TV (definition of midpoint)

the triangles are ≅ by SSS

Step 1: Given RS = TS, V is the midpoint of RT.

Step 2: Draw triangle RSV and triangle TSV, with RS and TS as the respective bases.

Step 3: Since V is the midpoint of RT, we can conclude that RV = VT by the definition of midpoint.

Step 4: RS = TS (Given).

Step 5: V is the midpoint of RT (Given).

Step 6: RV = VT (Step 3).

Step 7: Angle RSV = Angle TSV (Both are vertical angles).

Step 8: Triangle RSV is congruent to triangle TSV.

Step 9: Thus, we have proven /\RSV = /\TSV (By congruence of corresponding parts).

To write a two-column proof, we will list the statements on the left column and the corresponding reasons on the right column.

Statements Reasons
------------------------------------------------------
1. RS = TS Given
2. V is the midpoint of RT Given
3. RV = TV Definition of Midpoint
4. /\RSV ≅ /\TSV Side-Angle-Side (SAS) congruence theorem

Now, let's explain each step:

Statement 1: RS = TS (Given)
Reason: Given in the problem statement.

Statement 2: V is the midpoint of RT (Given)
Reason: Given in the problem statement.

Statement 3: RV = TV (Definition of Midpoint)
Reason: By definition, a midpoint divides a segment into two equal parts. Since V is the midpoint of RT, RV is equal to TV.

Statement 4: /\RSV ≅ /\TSV (Side-Angle-Side congruence theorem)
Reason: In order to prove that two triangles are congruent, we can use the Side-Angle-Side (SAS) congruence theorem. In this case, we have RS = TS (given), RV = TV (proved in statement 3), and the included angle SVR = SVT (common side). Therefore, /\RSV and /\TSV are congruent by SAS congruence theorem.

So, by listing the given statements, explaining the reasoning for each step, and indicating the congruence theorem used to prove congruence, a two-column proof for the given statement has been provided.

Actually, you have to use the midpoint theorem, not the definition of midpoint because you're talking about congruence statements, not equality.