Question

given: parallelogram lmno; \\( \overline{mo} \perp \overline{ln} \\) prove: lmno is a rhombus. statements reasons 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. \\( \diamond \\) 5. \\( \overline{np} \cong \overline{lp} \\) 5. diagonals of a parallelogram bisect each other 6. \\( \overline{mp} \cong \overline{mp} \\) 6. \\( \diamond \\) 7 \\( \triangle mnp \cong \triangle mlp \\) 8. sas 8. \\( \overline{lm} \cong \overline{mn} \\) 8. cpctc

Explanation:

Step1: Justify $\angle MPN\cong\angle MPL$

All right angles are congruent. Since $\angle MPN$ and $\angle MPL$ are right angles (from step 3), the reason for $\angle MPN\cong\angle MPL$ is "All right angles are congruent".

Step2: Justify $\overline{MP}\cong\overline{MP}$

A segment is congruent to itself. This is the reflexive property of congruence. So the reason for $\overline{MP}\cong\overline{MP}$ is "Reflexive property of congruence".

Step3: Complete the proof

Since we have shown $\triangle MNP\cong\triangle MLP$ (by SAS, with $\angle MPN\cong\angle MPL$, $\overline{NP}\cong\overline{LP}$, and $\overline{MP}\cong\overline{MP}$), and $\overline{LM}\cong\overline{MN}$ (by CPCTC). In a parallelogram, if adjacent sides are congruent, then the parallelogram is a rhombus. The reason for "LMNO is a rhombus" is "A parallelogram with adjacent sides congruent is a rhombus".

Answer:

1. All right angles are congruent; 6. Reflexive property of congruence; 9. A parallelogram with adjacent sides congruent is a rhombus.