Question

select the correct answer from each drop - down menu.
given: ( ux = wz ), ( vw = xy )
prove: ( uv = yz )
diagram: ( u---v---w---x---y---z )

statements	reasons
2. ( ux = uv + vw + wx ); ( wz = wx + xy + yz )	2. segment addition
3. ( uv + vw + wx = wx + xy + yz )	3. addition property of equality
4. ( uv + vw = xy + yz )	4. subtraction property of equality
5. ( vw = xy )	5. given
6. ( uv = yz )	6. sub

where is the first error in this proof?
the reason in line drop - down does not justify the statement because drop - down. instead, the reason should be drop - down.

Explanation:

To determine the first error in the proof, we analyze each step:

Step 1:

• Statement: \( UX = WZ \)
• Reason: Given (correct, as per the problem statement).

Step 2:

• Statement for \( UX \): \( UX = UV + VW + WX \) (correct, by segment addition postulate: \( U \to V \to W \to X \), so \( UX \) is the sum of \( UV \), \( VW \), and \( WX \)).
• Statement for \( WZ \): \( WZ = WX + XY + YZ \) (correct, by segment addition postulate: \( W \to X \to Y \to Z \), so \( WZ \) is the sum of \( WX \), \( XY \), and \( YZ \)).
• Reason: Segment addition (correct, as both use the segment addition postulate).

Step 3:

• Statement: \( UV + VW + WX = WX + XY + YZ \)
• Reason: Addition property of equality (correct, since \( UX = WZ \) (Step 1) and \( UX = UV + VW + WX \), \( WZ = WX + XY + YZ \) (Step 2), so substituting \( UX \) and \( WZ \) gives this equality).

Step 4:

• Statement: \( UV + VW = XY + YZ \)
• Reason: Subtraction property of equality (correct, subtracting \( WX \) from both sides of the equation in Step 3).

Step 5:

• Statement: \( VW = XY \)
• Reason: Given (correct, as per the problem statement \( VW = XY \)).

Step 6:

• Statement: \( UV = YZ \)
• Reason: Subtraction property of equality (Error Here). To get \( UV = YZ \), we should substitute \( VW = XY \) into \( UV + VW = XY + YZ \) (from Step 4) and simplify. The correct reason is the substitution property of equality (replace \( VW \) with \( XY \) in \( UV + VW = XY + YZ \), so \( UV + XY = XY + YZ \), then subtract \( XY \) from both sides). The reason listed as "Subtraction property" is incorrect because we first need to substitute \( VW = XY \) (from Step 5) into Step 4, not directly subtract.

The first error occurs in Step 6, where the reason "Subtraction property of equality" is incorrect. The correct reason should involve substituting \( VW = XY \) (from Step 5) into the equation from Step 4 (using the substitution property of equality) before subtracting.

The proof’s first error is in Step 6. The reason “Subtraction property of equality” is incorrect because to derive \( UV = YZ \), we must first substitute \( VW = XY \) (from Step 5) into \( UV + VW = XY + YZ \) (Step 4) (using the substitution property) and then subtract \( XY \) (subtraction property). The reason given in Step 6 does not account for the substitution of \( VW = XY \).

Answer:

The first error is in Step 6. The reason “Subtraction property of equality” is incorrect; the correct reasoning requires substituting \( VW = XY \) (from Step 5) into the equation from Step 4 (substitution property) before subtracting.