Question

1. given: \\(\overline{mn} \parallel \overline{po}\\), \\(\overline{mp} \parallel \overline{no}\\)

prove: \\(\overline{mp} \cong \overline{on}\\)
\\(\

$$\begin{array}{|c|c|} \\hline \\text{statements} & \\text{reasons} \\\\ \\hline 1. \\ \\overline{mn} \\parallel \\overline{po} & 1. \\text{given} \\\\ \\hline 2. \\ \\overline{mp} \\parallel \\overline{no} & 2. \\text{given} \\\\ \\hline 3. & 3. \\\\ \\hline 4. & 4. \\\\ \\hline 5. & 5. \\\\ \\hline 6. & 6. \\\\ \\hline 7. \\ \\overline{mp} \\cong \\overline{on} & 7. \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Identify the figure type

From the given parallel sides \(\overline{MN} \parallel \overline{PO}\) and \(\overline{MP} \parallel \overline{NO}\), we can conclude that quadrilateral \(MNOP\) is a parallelogram (by the definition of a parallelogram: a quadrilateral with both pairs of opposite sides parallel).

Step2: Recall properties of parallelograms

In a parallelogram, opposite sides are congruent. So in parallelogram \(MNOP\), \(\overline{MP}\) and \(\overline{ON}\) are opposite sides.

Step3: Fill in the proof table

1. Statement 1: \(\overline{MN} \parallel \overline{PO}\)

Reason 1: Given

1. Statement 2: \(\overline{MP} \parallel \overline{NO}\)

Reason 2: Given

1. Statement 3: Quadrilateral \(MNOP\) is a parallelogram

Reason 3: Definition of a parallelogram (both pairs of opposite sides parallel)

1. Statement 4: In a parallelogram, opposite sides are congruent

Reason 4: Property of parallelograms

1. Statement 5: \(\overline{MP}\) and \(\overline{ON}\) are opposite sides of parallelogram \(MNOP\)

Reason 5: Identification of opposite sides in \(MNOP\)

1. Statement 6: \(\overline{MP} \cong \overline{ON}\) (from step 4 and 5)

Reason 6: By the property of parallelograms (opposite sides congruent)

1. Statement 7: \(\overline{MP} \cong \overline{ON}\)

Reason 7: As proven in step 6

Answer:

The completed proof shows that \(\boldsymbol{\overline{MP} \cong \overline{ON}}\) using the properties of parallelograms. The key steps are recognizing the parallelogram from the given parallel sides and applying the opposite - sides - congruent property of parallelograms.