Question

proof:

statements	reasons
2. ?	2. definition of midpoint
3. ( sr = ur ) and ( rt = rv )	3. ?
4. ?	4. substitution

(diagram: horizontal line ( s---r---t ), vertical line ( c---r---v ), with ( r ) as the intersection point.)

Explanation:

Step1: Analyze Statement 1 Reason

From the diagram, \( R \) is the midpoint of \( ST \) (since \( S \), \( R \), \( T \) are colinear and \( R \) is between \( S \) and \( T \) with equal segments). So the reason for \( SR = RT \) is "Definition of midpoint (of \( ST \))".

Step2: Determine Statement 2

The reason is "Definition of midpoint", so the statement should be about \( R \) being the midpoint of another segment. Looking at the vertical segment \( UV \) (with \( U = C \) and \( V \)), the statement is \( CR = RV \) (or \( UR = RV \) where \( U = C \)).

Step3: Analyze Statement 3 Reason

We know \( SR = RT \) (from step 1) and \( SR = UR \), \( RT = RV \) (from step 2's midpoint definition). The reason for \( SR = UR \) and \( RT = RV \) is also "Definition of midpoint (of \( UV \))".

Step4: Determine Statement 4

Using substitution, since \( SR = RT \), \( SR = UR \), \( RT = RV \), we can substitute to get \( UR = RV \) (or \( CR = RV \) and \( SR = RT \) leads to \( UR = RV \)). But more precisely, from \( SR = RT \), \( SR = UR \), \( RT = RV \), substituting \( SR \) and \( RT \) gives \( UR = RV \).

Answer:

1. Reason: Definition of midpoint (of \( \overline{ST} \))
2. Statement: \( CR = RV \) (or \( UR = RV \))
3. Reason: Definition of midpoint (of \( \overline{UV} \))
4. Statement: \( UR = RV \) (or \( CR = RV \))

(Note: The exact labels for points \( C \) and \( V \) can be adjusted, but the key is using midpoint definitions and substitution. The main idea is recognizing midpoints and using substitution property.)