Question

proving a property of rhombi
given: parallelogram lmno; \\( \overline { mo } \perp \overline { ln } \\)
prove: lmno is a rhombus

1. lmno is a parallelogram 1. given
2. \\( \overline { mo } \perp \overline { ln } \\) 2. given
3. \\( \angle mpn \\) and \\( \angle mpl \\) are right angles 3. def. of perpendicular
4. \\( \angle mpn \cong \angle mpl \\) 4. •
5. \\( \overline { np } \cong \overline { lp } \\) 5. digonals of a parallelogram bisect each other
6. \\( mp \cong mp \\) 6. •

Explanation:

Step1: Right angles are congruent

Since \(\angle MPN\) and \(\angle MPL\) are right angles (from step 3), and all right angles are congruent, so \(\angle MPN\cong\angle MPL\) (reason for step 4).

Step2: Reflexive property of congruence

For \(MP\cong MP\) (step 6), by the reflexive property of congruence, a segment is congruent to itself.

Answer:

1. All right angles are congruent; 6. Reflexive property of congruence.