Question

vw ≅ yz, ∠vuz ≅ ∠wxy, and ∠uzv ≅ ∠wyx. complete the proof that xy ≅ uz.
image of a geometric figure with points x, w, y, v, z, u

statement	reason
2 ∠vuz ≅ ∠wxy
3 ∠uzv ≅ ∠wyx
4 vz = yz + vy
5 wy = vw + vy
6 vz = vw + vy
7 wy = vz
8 △unz ≅ △xwy
9 xy ≅ uz

Explanation:

Step1: Identify Given Information

We are given \( \overline{VW} \cong \overline{YZ} \), \( \angle VUZ \cong \angle WXY \), and \( \angle UZV \cong \angle WYX \). We need to prove \( \overline{XY} \cong \overline{UZ} \).

Step2: Analyze Step 1 (Statement: \( \overline{VW} \cong \overline{YZ} \))

The reason for this is "Given" (since it's provided in the problem statement).

Step3: Analyze Step 2 (Statement: \( \angle VUZ \cong \angle WXY \))

The reason is "Given" (as it's part of the problem's given information).

Step4: Analyze Step 3 (Statement: \( \angle UZV \cong \angle WYX \))

The reason is "Given" (from the problem's information).

Step5: Analyze Step 4 (Statement: \( VZ = YZ + VY \))

This is the "Segment Addition Postulate" (which states that if a point \( Y \) is on segment \( VZ \), then \( VZ = YZ + VY \)).

Step6: Analyze Step 5 (Statement: \( WY = VW + VY \))

This is also the "Segment Addition Postulate" (since \( V \) is on segment \( WY \), so \( WY = VW + VY \)).

Step7: Analyze Step 6 (Statement: \( VZ = VW + VY \))

We know \( \overline{VW} \cong \overline{YZ} \), so \( YZ = VW \) (Definition of Congruent Segments). Substitute \( YZ \) with \( VW \) in Step 4: \( VZ = VW + VY \). So the reason is "Substitution Property" (substituting \( YZ \) with \( VW \) because they are congruent).

Step8: Analyze Step 7 (Statement: \( WY = VZ \))

From Step 5 (\( WY = VW + VY \)) and Step 6 (\( VZ = VW + VY \)), by the "Transitive Property of Equality" (if \( a = b \) and \( b = c \), then \( a = c \)), we get \( WY = VZ \). So the reason is "Transitive Property of Equality".

Step9: Analyze Step 8 (Statement: \( \triangle U NZ \cong \triangle XWY \))

We have \( \angle VUZ \cong \angle WXY \) (Step 2), \( \angle UZV \cong \angle WYX \) (Step 3), and \( WY = VZ \) (Step 7). By the "AAS (Angle - Angle - Side) Congruence Criterion", \( \triangle U NZ \cong \triangle XWY \). AAS states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent. Here, \( \angle VUZ \cong \angle WXY \), \( \angle UZV \cong \angle WYX \), and \( VZ \cong WY \) (from Step 7, so the non - included sides are congruent), so AAS applies.

Step10: Analyze Step 9 (Statement: \( \overline{XY} \cong \overline{UZ} \))

Since \( \triangle U NZ \cong \triangle XWY \) (Step 8), by the "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)", the corresponding sides \( \overline{XY} \) and \( \overline{UZ} \) are congruent.

Now, filling in the reasons:

1. \( \overline{VW} \cong \overline{YZ} \): Given
2. \( \angle VUZ \cong \angle WXY \): Given
3. \( \angle UZV \cong \angle WYX \): Given
4. \( VZ = YZ + VY \): Segment Addition Postulate
5. \( WY = VW + VY \): Segment Addition Postulate
6. \( VZ = VW + VY \): Substitution Property (since \( YZ = VW \) from \( \overline{VW} \cong \overline{YZ} \))
7. \( WY = VZ \): Transitive Property of Equality (from \( WY = VW + VY \) and \( VZ = VW + VY \))
8. \( \triangle UNZ \cong \triangle XWY \): AAS (Angle - Angle - Side) Congruence Criterion (two angles and a non - included side congruent)
9. \( \overline{XY} \cong \overline{UZ} \): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

The completed proof with reasons is as follows:

Statement	Reason
2. \( \angle VUZ \cong \angle WXY \)	Given
3. \( \angle UZV \cong \angle WYX \)	Given
4. \( VZ = YZ + VY \)	Segment Addition Postulate
5. \( WY = VW + VY \)	Segment Addition Postulate
6. \( VZ = VW + VY \)	Substitution Property ( \( YZ = VW \) from \( \overline{VW} \cong \overline{YZ} \))
7. \( WY = VZ \)	Transitive Property of Equality
8. \( \triangle UNZ \cong \triangle XWY \)	AAS Congruence Criterion
9. \( \overline{XY} \cong \overline{UZ} \)	CPCTC