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
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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.