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 all right angles are congruent. reflexive property 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: Justify congruence of right - angles

Since $\angle MPN$ and $\angle MPL$ are right - angles (by definition of perpendicular lines), and all right - angles are congruent, so $\angle MPN\cong\angle MPL$.

Step2: Justify side equality

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

Step3: Prove triangle congruence

We know that $\overline{NP}\cong\overline{LP}$ (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, $\triangle MNP\cong\triangle MLP$.

Step4: Prove side equality in parallelogram

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

Answer:

The reasons for statements 4 and 6 are "All right angles are congruent" and "reflexive property" respectively. And since we have shown that two adjacent sides of the parallelogram $LMNO$ are congruent ($\overline{LM}\cong\overline{MN}$), parallelogram $LMNO$ is a rhombus.