Write a 2-column proof for each of the following

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

/\ is the sign that looks like a triangle.

Step 1: Write down the given information:

Given: RS =~ TS, V is the midpoint of RT

Step 2: Draw a diagram to represent the given information. The diagram should show triangle RSV and triangle TSV.

Step 3: Use the definition of triangle similarity to show that triangle RSV is similar to triangle TSV.

By the Side-Side-Side (SSS) similarity theorem, we need to show that the corresponding sides of triangle RSV and triangle TSV are proportional.

Step 4: Prove that RV =~ TV:

Since V is the midpoint of RT, we can conclude that RV =~ TV by the Midpoint Theorem, which states that a segment connecting the midpoint of a side of a triangle to the opposite vertex is half the length of the side.

Step 5: Prove that RS =~ TS:

Given that RS =~ TS, we don't need to prove it again.

Step 6: Conclude that triangle RSV is similar to triangle TSV:

Since RV =~ TV and RS =~ TS, we have two pairs of corresponding sides that are proportional. According to the SSS similarity theorem, triangle RSV is similar to triangle TSV.

Step 7: Write down the conclusion:

Therefore, by the definition of triangle similarity, /\RSV =~ /\TSV.

To prove that triangles RSV and TSV are congruent, we can use the concept of the midpoint and the side-angle-side (SAS) congruence theorem. Here is a 2-column proof for the given statements.

Statement | Reason
---------------------------------------------------------
1. RS = TS | Given
2. V is the midpoint of RT | Given
3. RV = TV | Definition of a midpoint
4. <RSV = <TSV | Vertical angles are congruent
5. /\RSV = /\TSV | SAS congruence theorem (sides RS = TS, angle RSV = TSV)

Explanation:
1. RS = TS: This statement is given in the problem.
2. V is the midpoint of RT: This statement is also given in the problem.
3. RV = TV: By the definition of a midpoint, when V is the midpoint of RT, both RV and TV are equal.
4. <RSV = <TSV: Vertical angles are two nonadjacent angles formed by intersecting lines. By definition, vertical angles are always congruent.
5. /\RSV = /\TSV: The triangles RSV and TSV have the same three corresponding parts (side, angle, side), satisfying the side-angle-side (SAS) congruence theorem. Therefore, the triangles are congruent.

Thus, the statement /\RSV = /\TSV is proved using the given information and the SAS congruence theorem.