do a two column for the following...
Given: line segment RV; length of line segment RT= length of line segment SV.
Prove: length of line segment RS= length of line segment TV.
what if S and T do not lie on RV?
r_____s________________t_______v
that is what the image looked like.
RS+ST+TV = RV
RT = RS+ST
SV = TV+ST
see what you can do with that.
To prove that the length of line segment RS is equal to the length of line segment TV, we can use the concept of congruence. If two line segments have the same length, they are considered congruent.
Here's a two-column proof to demonstrate the logic behind the proof:
| Statements | Reasons |
-------------------------------------------------
| 1. RT = SV | Given |
| 2. RT + RS = RV | Segment Addition Postulate |
| 3. SV + TV = RV | Segment Addition Postulate |
| 4. SV + RS = TV | Substitution (from 2) |
| 5. RT + RS = SV + RS | Substitution (from 1 and 4) |
| 6. RT = SV | Subtraction Property of Equality |
| 7. RT + RS = RT + RS | Addition Property of Equality |
| 8. RS = RS | Subtraction Property of Equality |
| 9. RT + RS = SV + RS | Transitive Property of Equality |
| 10. RS + RT = RS + SV | Commutative Property of Equality |
| 11. RS = SV | Subtraction Property of Equality |
| 12. RS = TV | Substitution (from 11) |
In this proof, we start with the given statement that line segment RT is equal in length to line segment SV. We then use the segment addition postulate to break down the line segment RV into its individual parts, leading to statements 2 and 3.
Next, we substitute the given information from statement 1 into statements 2 and 4. This allows us to combine the lengths of the line segments RS and SV to be equal to the length of TV.
In statement 5, we use the transitive property of equality to show that RT is equal to RS, which leads to statement 8, where RS is equal to itself.
In the subsequent statements, we simplify the equation and make use of properties of equality to show that RS is equal to TV, therefore proving the given statement.