Question

Given: MQ= NT, MQ||NT
Prove: MN = TQ

Answer 1

`\( MQ \cong NT \)` and
`\( MQ \parallel NT \), \( MQNT \)`is a parallelogram, and thus
`\( MN \cong TQ \)` by the properties of a parallelogram.

The image contains a geometric diagram with a quadrilateral
`\( MQNT \)` and a line segment
`\( MT \)`. The following information is given and required to be proved:

Given:

`\( MQ \cong NT \) (Segment \( MQ \) is congruent to segment \( NT \))`

`\( MQ \parallel NT \) (Line \( MQ \) is parallel to line \( NT \))`

Prove:

`\( MN \cong TQ \) (Segment \( MN \) is congruent to segment \( TQ \))`

The diagram shows quadrilateral
`\( MQNT \)` with
`\( MQ \)` and
`\( NT \)` as opposite sides and
`\( MN \)` and
`\( QT \)` as the other pair of opposite sides. There are arrows on lines
`\( MQ \) and \( NT \)` indicating they are parallel, and tick marks on
`\( MQ \) and \( NT \)` indicating they are of equal length.

To solve the given problem, we will use properties of parallelograms, as the given information suggests that
`\( MQNT \)` could be a parallelogram.

Given:
`\( MQ \cong NT \) and \( MQ \parallel NT \)`

To Prove:
`\( MN \cong TQ \)`

Step 1: Identify Properties

If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Step 2: Apply Properties

- By the definition of a parallelogram, the opposite sides are congruent. Therefore, if
`\( MQNT \)` is a parallelogram, then
`\( MN \cong TQ \)`.

Step 3: Conclusion

Since
`\( MQ \cong NT \)` and
`\( MQ \parallel NT \), \( MQNT \)`is a parallelogram, and thus
`\( MN \cong TQ \)` by the properties of a parallelogram.