Question

segment proofs
directions: complete the proof to the right using the statements and reasons.
given: ( 2wv = xy ); ( 2yz = xw ); ( overline{wv} cong overline{yz} )
prove: ( overline{xv} cong overline{xz} )
statements reasons

1. place statement here 1. place reason here
2. place statement here 2. place reason here
3. place statement here 3. place reason here
4. place statement here 4. place reason here
5. place statement here 5. place reason here
6. place statement here 6. place reason here
7. place statement here 7. place reason here
8. place statement here 8. place reason here

not all reasons will be used!
definition of congruence
definition of congruence
definition of midpoint
given
multiplication property
segment addition postulate
segment addition postulate
substitution property
substitution property
transitive property
subtraction property

Explanation:

Step1: List Given Statements

1. \( 2WV = XY \); \( 2YZ = XW \); \( \overline{WV} \cong \overline{YZ} \)

Reason: Given

Step2: Use Definition of Congruence

1. \( WV = YZ \)

Reason: Definition of Congruence (since \( \overline{WV} \cong \overline{YZ} \), their lengths are equal)

Step3: Multiply Both Sides by 2

1. \( 2WV = 2YZ \)

Reason: Multiplication Property (multiplying both sides of \( WV = YZ \) by 2)

Step4: Substitute Given Equalities

1. \( XY = XW \)

Reason: Substitution Property (substitute \( 2WV = XY \) and \( 2YZ = XW \) into \( 2WV = 2YZ \))

Step5: Apply Segment Addition Postulate

1. \( XW + WV = XV \); \( XY + YZ = XZ \)

Reason: Segment Addition Postulate (for segments \( XW, WV \) forming \( XV \) and \( XY, YZ \) forming \( XZ \))

Step6: Substitute \( XY = XW \) and \( WV = YZ \)

1. \( XY + YZ = XV \)

Reason: Substitution Property (substitute \( XW = XY \) and \( WV = YZ \) into \( XW + WV = XV \))

Step7: Set \( XV \) and \( XZ \) Equal

1. \( XV = XZ \)

Reason: Substitution Property (since \( XY + YZ = XV \) and \( XY + YZ = XZ \), so \( XV = XZ \))

Step8: Conclude Congruence

1. \( \overline{XV} \cong \overline{XZ} \)

Reason: Definition of Congruence (if \( XV = XZ \), then their segments are congruent)

Answer:

The completed proof has the following statements and reasons:

1. \( 2WV = XY \); \( 2YZ = XW \); \( \overline{WV} \cong \overline{YZ} \) - Given
2. \( WV = YZ \) - Definition of Congruence
3. \( 2WV = 2YZ \) - Multiplication Property
4. \( XY = XW \) - Substitution Property
5. \( XW + WV = XV \); \( XY + YZ = XZ \) - Segment Addition Postulate
6. \( XY + YZ = XV \) - Substitution Property
7. \( XV = XZ \) - Substitution Property
8. \( \overline{XV} \cong \overline{XZ} \) - Definition of Congruence

And the final conclusion is \( \overline{XV} \cong \overline{XZ} \).