Question

given: parallelogram lmno; mo ⊥ ln prove: lmno is a rhombus.
statements reasons

1. lmno is a parallelogram 1. given
2. mo ⊥ ln 2. given
3. ∠mpn and ∠mpl are right angles 3. def of perpendicular
4. ∠mpn ≅ ∠mpl 4. *
5. np ≅ lp 5. diagonals of a parallelogram bisect each other
6. mp ≅ mp 6. *
7. △mnp ≅ △mlp 7. sas
8. lm ≅ mn 8. cpctc

opposite angles of a parallelogram are congruent. opposite sides of a parallelogram are congruent. diagonals of a parallelogram are congruent. diagonals of a parallelogram bisect each other

Explanation:

Step1: Define right - angle congruence

All right angles are congruent. So, since $\angle MPN$ and $\angle MPL$ are right angles (from step 3), $\angle MPN\cong\angle MPL$ because right - angle congruence.

Step2: Reflexive property

The side $\overline{MP}$ is common to both $\triangle MNP$ and $\triangle MLP$. So, $\overline{MP}\cong\overline{MP}$ by the reflexive property of congruence.

Step3: Complete the proof

We have $\overline{NP}\cong\overline{LP}$ (diagonals of a parallelogram bisect each other), $\angle MPN\cong\angle MPL$, and $\overline{MP}\cong\overline{MP}$. By the Side - Angle - Side (SAS) congruence criterion, $\triangle MNP\cong\triangle MLP$. Then, by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), $\overline{LM}\cong\overline{MN}$. Since $LMNO$ is a parallelogram and a pair of adjacent sides are congruent, $LMNO$ is a rhombus.

Answer:

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