Question

3.

Reasons
Statements
MNOP is a parallelogram
N
PM || ON
M
Given:
Alternate Int. Zs Thm,
MNOP is a parallelogram
MN II PO
Alternate Int. Zs Thm,
Prove:
PM = ON
(For this proof, use only the
definition of a parallelogram;
don't use any properties)
HE

Answer 1

3.
`\overline{PM}\cong \overline{ON}` Distances between two parallel lines
`\overline{MN} \ and\ \overline{PO}`

4.
`\overline{AC}` =
`\overline{CE}`: Reason; Corresponding part of ΔACB and ΔDCE

C is the midpoint of
`\overline{AE}`: Reason;
`\overline{AC}` =
`\overline{CE}`: Definition of midpoint

Explanation:

3. A parallelogram is defined as a quadrilateral with two opposite sides equal and parallel and having equal opposite interior angles

MNOP is a parallelogram: Reason; Given

`\overline{PM}\left | \right |\overline{ON}` : Reason; Opposite sides of a parallelogram

∠NOM ≅ ∠OMP: Reason Alternate interior angles

`\overline{MN}\left | \right |\overline{PO}`: Reason; Opposite sides of a parallelogram

∠NMO ≅ ∠MOP: Reason Alternate interior angles

`\overline{PM}\cong \overline{ON}` Distances between two parallel lines
`\overline{MN} \ and\ \overline{PO}`

4.
`\overline{AB}\left | \right |\overline{DE}` : Reason; Given

∠EAB ≅ ∠AED: Reason; Alternate int. ∠s Thm

∠ABC ≅ ∠EDB : Reason; Alternate int. ∠s Thm

C is the midpoint of
`\overline{BD}`: Reason; Given

`\overline{BC}` =
`\overline{CD}`: Reason; Definition of midpoint

Therefore, ΔACB ≅ ΔDCE: Reason Angle Angle Side (AAS) Theorem

`\overline{AC}` =
`\overline{CE}`: Reason; Corresponding part of ΔACB and ΔDCE

C is the midpoint of
`\overline{AE}`: Reason; Definition of midpoint