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 8. sas 8. lm ≅ mn 8. cpctc

Explanation:

Step1: Define right - angle congruence

All right angles are congruent. So, since $\angle MPN$ and $\angle MPL$ are right angles, $\angle MPN\cong\angle MPL$ because all right angles have a measure of $90^{\circ}$ and angles with the same measure are congruent.

Step2: Reflexive property of congruence

$\overline{MP}\cong\overline{MP}$ by the reflexive property of congruence, which states that any segment is congruent to itself.

Step3: Prove triangle congruence

In $\triangle MNP$ and $\triangle MLP$, we have $\angle MPN\cong\angle MPL$ (from step 1), $\overline{NP}\cong\overline{LP}$ (diagonals of a parallelogram bisect each other), and $\overline{MP}\cong\overline{MP}$ (from step 2). By the Side - Angle - Side (SAS) congruence criterion, $\triangle MNP\cong\triangle MLP$.

Step4: Use CPCTC

Since $\triangle MNP\cong\triangle MLP$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, $\overline{LM}\cong\overline{MN}$. In a parallelogram, if two adjacent sides are congruent, then the parallelogram is a rhombus.

Answer:

The reasons for statements 4 and 6 are:

1. All right angles are congruent.
2. Reflexive property of congruence.