complete the proof that $\overline{uv} \cong \overline{wx}$.

| | statement | reason |
| 1 | $\angle uzv \cong \angle wyx$ | |
| 2 | $\overline{vw} \cong \overline{yz}$ | |
| 3 | $\angle xwy \cong \angle uvz$ | |
| 4 | $vz = yz + vy$ | |
| 5 | $wy = vw + vy$ | |
| 6 | $vz = vw + vy$ | |
| 7 | $wy = vz$ | |
| 8 | $\triangle uvz \cong \triangle xwy$ | |
| 9 | $\overline{uv} \cong \overline{wx}$ | |

Question

complete the proof that $\overline{uv} \cong \overline{wx}$.

| | statement | reason |
| 1 | $\angle uzv \cong \angle wyx$ |  |
| 2 | $\overline{vw} \cong \overline{yz}$ |  |
| 3 | $\angle xwy \cong \angle uvz$ |  |
| 4 | $vz = yz + vy$ |  |
| 5 | $wy = vw + vy$ |  |
| 6 | $vz = vw + vy$ |  |
| 7 | $wy = vz$ |  |
| 8 | $\triangle uvz \cong \triangle xwy$ |  |
| 9 | $\overline{uv} \cong \overline{wx}$ |  |

Accepted Answer

# Explanation:
## Step1: Identify the reason for ∠UZV ≅ ∠WYX
From the diagram, ∠UZV and ∠WYX are marked with the same arc, so they are congruent by the "Given" (or "Marked Angles are Congruent") reason.
## Step2: Reason for $\overline{VW} \cong \overline{YZ}$
Looking at the diagram, $\overline{VW}$ and $\overline{YZ}$ have tick marks, so they are congruent by "Given" (or "Marked Segments are Congruent").
## Step3: Reason for ∠XWY ≅ ∠UVZ
These angles are marked with the same arc (or from the diagram's markings), so "Given" (or "Marked Angles are Congruent").
## Step4: Reason for $VZ = YZ + VY$
This is the Segment Addition Postulate, which states that if a point $Y$ is between $V$ and $Z$, then $VZ = VY + YZ$ (or $YZ + VY$).
## Step5: Reason for $WY = VW + VY$
By the Segment Addition Postulate, since $V$ is between $W$ and $Y$, $WY = VW + VY$.
## Step6: Reason for $VZ = VW + VY$
From Step 2, $\overline{VW} \cong \overline{YZ}$ implies $VW = YZ$. From Step 4, $VZ = YZ + VY$, so substituting $YZ$ with $VW$ (since $VW = YZ$) gives $VZ = VW + VY$ (Substitution Property).
## Step7: Reason for $WY = VZ$
From Step 5, $WY = VW + VY$, and from Step 6, $VZ = VW + VY$, so by the Transitive Property (or Substitution), $WY = VZ$.
## Step8: Reason for $\triangle UVZ \cong \triangle XWY$
We have $\angle UZV \cong \angle WYX$ (Step 1), $\angle XWY \cong \angle UVZ$ (Step 3), and $WY = VZ$ (Step 7), and $\overline{VW} \cong \overline{YZ}$ (Step 2, which can be part of the ASA or AAS congruence). Wait, actually, with $\angle UZV \cong \angle WYX$, $\angle UVZ \cong \angle XWY$, and $VZ = WY$ (Step 7), this is AAS (Angle - Angle - Side) congruence criterion. So $\triangle UVZ \cong \triangle XWY$ by AAS.
## Step9: Reason for $\overline{UV} \cong \overline{WX}$
Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Since $\triangle UVZ \cong \triangle XWY$, their corresponding sides $\overline{UV}$ and $\overline{WX}$ are congruent.

# Answer:
1. Given (or Marked Angles are Congruent)
2. Given (or Marked Segments are Congruent)
3. Given (or Marked Angles are Congruent)
4. Segment Addition Postulate
5. Segment Addition Postulate
6. Substitution Property (using $VW = YZ$ from Step 2)
7. Transitive Property (or Substitution, since $WY = VW + VY$ and $VZ = VW + VY$)
8. AAS (Angle - Angle - Side) Congruence Criterion
9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

	statement	reason
1	\\(\angle uzv \cong \angle wyx\\)
2	\\(\overline{vw} \cong \overline{yz}\\)
3	\\(\angle xwy \cong \angle uvz\\)
4	\\(vz = yz + vy\\)
5	\\(wy = vw + vy\\)
6	\\(vz = vw + vy\\)
7	\\(wy = vz\\)
8	\\(\triangle uvz \cong \triangle xwy\\)
9	\\(\overline{uv} \cong \overline{wx}\\)

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the reason for ∠UZV ≅ ∠WYX

Step2: Reason for \(\overline{VW} \cong \overline{YZ}\)

Step3: Reason for ∠XWY ≅ ∠UVZ

Step4: Reason for \(VZ = YZ + VY\)

Step5: Reason for \(WY = VW + VY\)

Step6: Reason for \(VZ = VW + VY\)

Step7: Reason for \(WY = VZ\)

Step8: Reason for \(\triangle UVZ \cong \triangle XWY\)

Step9: Reason for \(\overline{UV} \cong \overline{WX}\)

Answer: