Question

$\overline{uw} \cong \overline{vx}$. complete the proof that $\overline{vw} \perp \overline{uv}$.

1. $\overline{uw} \cong \overline{vx}$
2. $\overline{wx} \parallel \overline{uv}$
3. $\overline{vw} \parallel \overline{ux}$
4. $\overline{uv} \cong \overline{wx}$
5. $\overline{vw} \cong \overline{vw}$
6. $\triangle uvw \cong \triangle xwv$
7. $\angle uvw \cong \angle vwx$
8. $m\angle uvw + m\angle vwx = 180^\circ$
9. $m\angle uvw + m\angle uvw = 180^\circ$
10. $m\angle uvw = 90^\circ$
11. $\overline{vw} \perp \overline{uv}$

Explanation:

To complete the proof that \(\overline{VW} \perp \overline{UV}\), we analyze each statement and provide the corresponding reason:

Step 1: Statement \(\overline{UW} \cong \overline{VX}\)

• Reason: Given (as stated in the problem).

Step 2: Statement \(\overline{WX} \parallel \overline{UV}\)

• Reason: Opposite sides of a parallelogram are parallel (from the diagram, \(UVWX\) appears to be a parallelogram with marked parallel sides).

Step 3: Statement \(\overline{VW} \parallel \overline{UX}\)

• Reason: Opposite sides of a parallelogram are parallel (consistent with the parallelogram structure).

Step 4: Statement \(\overline{UV} \cong \overline{WX}\)

• Reason: Opposite sides of a parallelogram are congruent.

Step 5: Statement \(\overline{VW} \cong \overline{VW}\)

• Reason: Reflexive property of congruence (a segment is congruent to itself).

Step 6: Statement \(\triangle UVW \cong \triangle XWV\)

• Reason: SSS (Side - Side - Side) Congruence Postulate. We have \(\overline{UW} \cong \overline{VX}\) (Step 1), \(\overline{UV} \cong \overline{WX}\) (Step 4), and \(\overline{VW} \cong \overline{VW}\) (Step 5).

Step 7: Statement \(\angle UVW \cong \angle VWX\)

• Reason: Corresponding parts of congruent triangles are congruent (CPCTC). Since \(\triangle UVW \cong \triangle XWV\), their corresponding angles are congruent.

Step 8: Statement \(m\angle UVW + m\angle VWX = 180^\circ\)

• Reason: Consecutive interior angles are supplementary (because \(\overline{UV} \parallel \overline{WX}\) and \(\overline{VW}\) is a transversal).

Step 9: Statement \(m\angle UVW + m\angle UVW = 180^\circ\)

• Reason: Substitution (since \(\angle UVW \cong \angle VWX\), we can substitute \(m\angle VWX\) with \(m\angle UVW\)).

Step 10: Statement \(m\angle UVW = 90^\circ\)

• Reason: Solving the equation \(2m\angle UVW=180^\circ\) (from Step 9: \(m\angle UVW + m\angle UVW=180^\circ\) or \(2m\angle UVW = 180^\circ\)) gives \(m\angle UVW=\frac{180^\circ}{2} = 90^\circ\).

Step 11: Statement \(\overline{VW} \perp \overline{UV}\)

• Reason: If the measure of an angle between two segments is \(90^\circ\), then the segments are perpendicular. Since \(m\angle UVW = 90^\circ\), \(\overline{VW}\) and \(\overline{UV}\) form a right angle, so \(\overline{VW} \perp \overline{UV}\).

For the first statement (\(\overline{UW} \cong \overline{VX}\)) the reason is "Given".

If we are only asked for the reason of the first statement:

The statement \(\overline{UW} \cong \overline{VX}\) is given in the problem, so the reason is "Given".

Answer:

Given