Question

$\angle twu \cong \angle rvx$, $\overline{uv} \cong \overline{wx}$, and $\angle t \cong \angle r$. complete the proof that $\overline{tu} \cong \overline{rx}$.

(image of two triangles with points t, u, v, w, s, r, x)

statement	reason
2 $\overline{uv} \cong \overline{wx}$
3 $\angle t \cong \angle r$
4 $vx = wx + vw$
5 $uw = uv + vw$
6 $vx = uv + vw$
7 $uw = vx$
8 $\triangle rvx \cong \triangle twu$
9 $\overline{tu} \cong \overline{rx}$

Explanation:

Step1: Identify Given Information

We are given \(\angle TWU \cong \angle RVX\) (Statement 1), \(\overline{UV} \cong \overline{WX}\) (Statement 2), and \(\angle T \cong \angle R\) (Statement 3). We need to prove \(\overline{TU} \cong \overline{RX}\) by showing \(\triangle RVX \cong \triangle TWU\) and using corresponding parts of congruent triangles.

Step2: Analyze Segment Addition

For Statement 4 (\(VX = WX + VW\)) and Statement 5 (\(UW = UV + VW\)), the reason is the Segment Addition Postulate (a segment is the sum of its parts). Since \(\overline{UV} \cong \overline{WX}\) (Statement 2), substitute \(WX\) with \(UV\) in Statement 4: \(VX = UV + VW\) (Statement 6, Substitution Property). Then, \(UW = VX\) (Statement 7, Transitive Property, as both equal \(UV + VW\)).

Step3: Prove Triangle Congruence

Now, in \(\triangle RVX\) and \(\triangle TWU\):

• \(\angle TWU \cong \angle RVX\) (Given, Statement 1)
• \(UW \cong VX\) (Proven in Statement 7)
• \(\angle T \cong \angle R\) (Given, Statement 3)

This satisfies the AAS (Angle - Angle - Side) congruence criterion. So, \(\triangle RVX \cong \triangle TWU\) (Statement 8, AAS Congruence).

Step4: Corresponding Parts of Congruent Triangles

By the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, since \(\triangle RVX \cong \triangle TWU\), their corresponding sides \(\overline{TU}\) and \(\overline{RX}\) are congruent (Statement 9).

Filling in Reasons:

1. \(\angle TWU \cong \angle RVX\): Given
2. \(\overline{UV} \cong \overline{WX}\): Given
3. \(\angle T \cong \angle R\): Given
4. \(VX = WX + VW\): Segment Addition Postulate
5. \(UW = UV + VW\): Segment Addition Postulate
6. \(VX = UV + VW\): Substitution Property (from Statement 2: \(WX = UV\))
7. \(UW = VX\): Transitive Property (from Statements 5 and 6)
8. \(\triangle RVX \cong \triangle TWU\): AAS (Angle - Angle - Side) Congruence (using \(\angle TWU \cong \angle RVX\), \(UW \cong VX\), \(\angle T \cong \angle R\))
9. \(\overline{TU} \cong \overline{RX}\): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

The reasons for each statement are:

1. Given
2. Given
3. Given
4. Segment Addition Postulate
5. Segment Addition Postulate
6. Substitution Property
7. Transitive Property
8. AAS Congruence
9. CPCTC

(And thus \(\overline{TU} \cong \overline{RX}\) is proven.)