Question

$$overline{uv} cong overline{wx}, overline{yz} cong overline{vw}, \text{ and } overline{xy} cong overline{uz}. \text{ complete the proof that } angle x cong angle u.$$

statement	reason
2 $overline{yz} cong overline{vw}$	given
3 $overline{xy} cong overline{uz}$	given
4 $vz = yz + wy$	additive property of length
5 $wy = vw + wy$	additive property of length
6 $vz = vw + wy$	substitution
7 $wy = vz$	transitive property of equality
8

Explanation:

Step1: Analyze Triangle Congruence

We aim to prove \( \angle X \cong \angle U \). First, we have sides: \( \overline{UV} \cong \overline{WX} \) (Given), \( \overline{YZ} \cong \overline{VW} \) (Given), \( \overline{XY} \cong \overline{UZ} \) (Given). We need to show \( \triangle WXZ \cong \triangle UVZ \) or related triangles. From the statements, we build \( XZ \) and \( UZ \) lengths.

Step2: Complete the Proof Steps

Looking at the table, step 8 is missing. We know from step 7 \( WY = VZ \), and we have sides to apply SSS (Side - Side - Side) congruence. Then, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \angle X \cong \angle U \). The missing statement for step 8 should be \( \triangle WXZ \cong \triangle UVZ \) (SSS) and the reason "SSS Congruence Postulate", then step 9 (if needed) would be \( \angle X \cong \angle U \) by CPCTC. But focusing on the table, the last step (step 8) likely involves showing triangle congruence.

Step3: Finalize the Proof

After establishing all sides equal (from the additive and substitution properties), we use SSS to prove the triangles congruent, then CPCTC for the angles. So the missing statement in step 8 (assuming the table is building to SSS) would be \( \overline{XZ} \cong \overline{UZ} \) (wait, no, we already have \( \overline{XY} \cong \overline{UZ} \), \( \overline{WX} \cong \overline{UV} \), and we need \( \overline{XZ} \cong \overline{UZ} \)? No, better: we have \( WX = UV \), \( XY = UZ \), and \( XZ = WX + XY \)? Wait, no, the segments: \( VZ = WY \), \( WY = VW + WY \)? Wait, maybe the correct approach is:

From step 7: \( WY = VZ \) (Transitive Property). We have \( WX = UV \) (Given), \( VW = YZ \) (Given), so \( WX + XY = UV + YZ \)? No, let's re - express. The triangles are \( \triangle WXZ \) and \( \triangle UVZ \). Sides: \( WX = UV \) (Given), \( XY = UZ \) (Given), and \( XZ = WX + XY \)? No, \( XZ \) is composed of \( WX \) and \( XY \)? Wait, the diagram shows \( X - W - V - U - Z \)? No, the diagram has \( X, W \) and \( U, V \) with \( Y, Z \). So \( \triangle WXZ \) and \( \triangle UVZ \): \( WX = UV \) (Given), \( XY = UZ \) (Given), and \( XZ = WX + XY \), \( UZ = UV + VZ \), but we proved \( WY = VZ \), and \( VW = YZ \), so \( XZ = WX + XY \), \( UZ = UV + VZ \), and since \( WX = UV \), \( XY = UZ \), and \( VZ = WY \), but maybe the correct congruence is SSS with \( WX = UV \), \( XY = UZ \), and \( XZ = UZ \)? No, I think the key is that after step 7, we can say \( \triangle WXZ \cong \triangle UVZ \) by SSS (since \( WX = UV \), \( XY = UZ \), and \( XZ = UZ \)? No, better: the sides are \( WX = UV \), \( VW = YZ \), and \( XZ = UZ \) (no, I'm overcomplicating). The standard proof for this is:

1. \( \overline{UV} \cong \overline{WX} \) (Given)
2. \( \overline{YZ} \cong \overline{VW} \) (Given)
3. \( \overline{XY} \cong \overline{UZ} \) (Given)
4. \( VZ = YZ + WY \) (Additive Property of Length)
5. \( WY = VW + WY \)? No, step 4: \( VZ = YZ + WY \) (Additive Property, since \( Y \) is between \( W \) and \( V \)? Wait, the diagram: \( X - W \), \( U - V \), \( Y \) connected to \( Z \), \( W \), \( U \). So \( WY \) is a segment, \( VZ \) is a segment. After using additive and substitution, we get \( WY = VZ \), then \( WX + XY = UV + YZ \)? No, the correct final step before CPCTC is proving the triangles congruent by SSS. So the missing statement in step 8 (the last step in the table) should be \( \triangle WXZ \cong \triangle UVZ \) (SSS) and the reason "SSS Congruence Postulate", then the next step (step 9) would be \( \angle X \cong \angle U \) by CPCTC.

Answer:

The missing statement for step 8 (to complete the proof) is \( \triangle WXZ \cong \triangle UVZ \) (by SSS Congruence Postulate), and then \( \angle X \cong \angle U \) (by CPCTC). If we consider the table's structure, the final answer for the proof of \( \angle X \cong \angle U \) is established by proving \( \triangle WXZ \cong \triangle UVZ \) (SSS) and then using CPCTC, so the conclusion is \( \angle X \cong \angle U \).