Question

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

(image of a quadrilateral with vertices v, w, x, u and diagonals intersecting at y, with some sides marked as parallel or congruent)

	statement	reason
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	$mangle uvw + mangle vwx = 180^circ$
9	$mangle uvw + mangle uvw = 180^circ$
10	$mangle uvw = 90^circ$
11	$overline{vw} perp overline{uv}$

Explanation:

Step1: Identify Given Information

The first statement is \(\overline{UW} \cong \overline{VX}\), and this is given in the problem. So the reason for this statement is "Given".

Step2: Analyze Parallel Sides (Statement 2)

Looking at the diagram, \(\overline{WX}\) and \(\overline{UV}\) have the same number of tick marks, indicating they are congruent and parallel (since in a parallelogram - like figure, opposite sides are parallel). So the reason for \(\overline{WX} \parallel \overline{UV}\) is "Opposite sides of a parallelogram are parallel" (assuming the figure is a parallelogram from the tick marks on sides).

Step3: Analyze Parallel Sides (Statement 3)

Similarly, \(\overline{VW}\) and \(\overline{UX}\) have tick marks, so \(\overline{VW} \parallel \overline{UX}\) with the reason "Opposite sides of a parallelogram are parallel".

Step4: Congruent Opposite Sides (Statement 4)

\(\overline{UV} \cong \overline{WX}\) because opposite sides of a parallelogram are congruent, so the reason is "Opposite sides of a parallelogram are congruent".

Step5: Reflexive Property (Statement 5)

\(\overline{VW} \cong \overline{VW}\) by the reflexive property of congruence (a segment is congruent to itself), so the reason is "Reflexive property of congruence".

Step6: SSS Congruence (Statement 6)

We have \(\overline{UW} \cong \overline{VX}\), \(\overline{UV} \cong \overline{WX}\), and \(\overline{VW} \cong \overline{VW}\), so by SSS (Side - Side - Side) congruence criterion, \(\triangle UVW \cong \triangle XWV\). Reason: "SSS Congruence Criterion".

Step7: Corresponding Parts (Statement 7)

Since \(\triangle UVW \cong \triangle XWV\), their corresponding angles \(\angle UVW\) and \(\angle VWX\) are congruent. Reason: "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Step8: Linear Pair (Statement 8)

\(\angle UVW\) and \(\angle VWX\) form a linear pair (they are adjacent and form a straight line), so \(m\angle UVW + m\angle VWX = 180^{\circ}\). Reason: "Linear Pair Postulate".

Step9: Substitution (Statement 9)

Since \(\angle UVW \cong \angle VWX\), we can substitute \(m\angle VWX\) with \(m\angle UVW\) in the equation \(m\angle UVW + m\angle VWX = 180^{\circ}\), giving \(m\angle UVW + m\angle UVW = 180^{\circ}\). Reason: "Substitution Property".

Step10: Solve for Angle Measure (Statement 10)

Simplify \(m\angle UVW + m\angle UVW = 180^{\circ}\) to \(2m\angle UVW = 180^{\circ}\), then divide both sides by 2: \(m\angle UVW=\frac{180^{\circ}}{2} = 90^{\circ}\). Reason: "Division Property of Equality".

Step11: Definition of Perpendicular (Statement 11)

If the measure of an angle between two segments is \(90^{\circ}\), then the segments are perpendicular. So since \(m\angle UVW = 90^{\circ}\), \(\overline{VW} \perp \overline{UV}\). Reason: "Definition of Perpendicular Lines".

Answer:

1. Given
2. Opposite sides of a parallelogram are parallel
3. Opposite sides of a parallelogram are parallel
4. Opposite sides of a parallelogram are congruent
5. Reflexive property of congruence
6. SSS Congruence Criterion
7. CPCTC
8. Linear Pair Postulate
9. Substitution Property
10. Division Property of Equality
11. Definition of Perpendicular Lines