Question

proof
4
segment proofs
directions: complete the proof to the right using the statements and reasons.
given: n is the midpoint of (overline{jl});
n is the midpoint of (overline{km});
(jl = km)
prove: (kn = nl)
not all reasons will be used!
definition of congruence
definition of midpoint
definition of midpoint
given
division property
segment addition postulate
segment addition postulate
simplify
substitution property
substitution property
transitive property

Explanation:

Step1: Identify Given Information

The given information is: \( N \) is the midpoint of \( \overline{JL} \); \( N \) is the midpoint of \( \overline{KM} \); \( JL = KM \). So the first statement should be these given facts, and the reason is "Given".
Statement 1: \( N \) is the midpoint of \( \overline{JL} \); \( N \) is the midpoint of \( \overline{KM} \); \( JL = KM \)
Reason 1: Given

Step2: Apply Definition of Midpoint

By the definition of midpoint, if \( N \) is the midpoint of \( \overline{JL} \), then \( JN = NL \). If \( N \) is the midpoint of \( \overline{KM} \), then \( KN = NM \).
Statement 2: \( JN = NL \); \( KN = NM \)
Reason 2: Definition of Midpoint

Step3: Apply Segment Addition Postulate

The Segment Addition Postulate states that for a segment, the sum of its parts equals the whole. For \( \overline{JL} \), \( JN + NL = JL \). For \( \overline{KM} \), \( KN + NM = KM \).
Statement 3: \( JN + NL = JL \); \( KN + NM = KM \)
Reason 3: Segment Addition Postulate

Step4: Substitute \( JL \) and \( KM \)

We know \( JL = KM \) (given), so substitute \( JL \) with \( KM \) in the equations from Step 3. So \( JN + NL = KN + NM \).
Statement 4: \( JN + NL = KN + NM \)
Reason 4: Substitution Property (since \( JL = KM \))

Step5: Substitute \( JN \) and \( NM \)

From Step 2, we know \( JN = NL \) and \( KN = NM \). Substitute \( JN \) with \( NL \) and \( NM \) with \( KN \) in the equation from Step 4. So \( NL + NL = KN + KN \), which simplifies to \( 2NL = 2KN \) or \( KN + KN = NL + NL \).
Statement 5: \( KN + KN = NL + NL \) (or \( 2KN = 2NL \))
Reason 5: Substitution Property (substituting \( JN = NL \) and \( KN = NM \))

Step6: Simplify the Equation

Divide both sides of \( 2KN = 2NL \) by 2 (Division Property) or simplify \( KN + KN = NL + NL \) to \( 2KN = 2NL \) and then to \( KN = NL \).
Statement 6: \( 2KN = 2NL \) (from simplifying \( KN + KN = NL + NL \))
Reason 6: Simplify (or Division Property, dividing both sides by 2)

Step7: Conclude \( KN = NL \)

By dividing both sides of \( 2KN = 2NL \) by 2, we get \( KN = NL \).
Statement 7: \( KN = NL \)
Reason 7: Division Property (or Simplify, as dividing by 2 simplifies the equation)

Answer:

The completed proof is as follows:

Statements	Reasons
2. \( JN = NL \); \( KN = NM \)	2. Definition of Midpoint
3. \( JN + NL = JL \); \( KN + NM = KM \)	3. Segment Addition Postulate
4. \( JN + NL = KN + NM \)	4. Substitution Property (since \( JL = KM \))
5. \( KN + KN = NL + NL \) (or \( 2KN = 2NL \))	5. Substitution Property (substituting \( JN = NL \), \( KN = NM \))
6. \( 2KN = 2NL \) (or simplified from Step 5)	6. Simplify (or Division Property)
7. \( KN = NL \)	7. Division Property (dividing both sides by 2)

The final conclusion is \( \boldsymbol{KN = NL} \).