Question

given: quadrilateral mnol with \\(\overline{mn} \cong \overline{lo}\\) and \\(\overline{ml} \cong \overline{no}\\) prove: mnol is a parallelogram. complete the paragraph proof. we are given that \\(\overline{mn} \cong \overline{lo}\\) and \\(\overline{ml} \cong \overline{no}\\). we can draw in \\(\overline{mo}\\) because between any two points is a line. by the reflexive property, \\(\overline{mo} \cong \overline{mo}\\). by sss, \\(\triangle mlo \cong \triangle \square\\). by cpctc, \\(\angle lmo \cong \angle \square\\) and \\(\angle nmo \cong \angle lom\\). both pairs of angles are also \\(\square\\), based on the definition. based on the converse of the alternate interior angles theorem, \\(\overline{mn} \parallel \overline{lo}\\) and \\(\overline{lm} \parallel \overline{no}\\). based on the definition of a parallelogram, mnol is a parallelogram.

Explanation:

Step-by-Step Explanation:

Step 1: Identify the congruent triangles

We know \( \overline{MN} \cong \overline{LO} \), \( \overline{ML} \cong \overline{NO} \), and \( \overline{MO} \cong \overline{MO} \) (reflexive property). By SSS (Side - Side - Side) congruence criterion, \( \triangle MLO \cong \triangle NOM \).

Step 2: Apply CPCTC for the first angle

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the corresponding angle to \( \angle LMO \) in \( \triangle NOM \) is \( \angle NOM \). So \( \angle LMO \cong \angle NOM \).

Step 3: Identify the type of angles

The angles \( \angle LMO \) and \( \angle NOM \), \( \angle NMO \) and \( \angle LOM \) are alternate interior angles. This is because they lie between the two lines (the sides of the quadrilateral) and on opposite sides of the transversal \( \overline{MO} \).

Answer:

s for the blanks:

1. The triangle congruent to \( \triangle MLO \) is \( \triangle NOM \).
2. The angle congruent to \( \angle LMO \) is \( \angle NOM \).
3. The type of angles ( \( \angle LMO \) and \( \angle NOM \), \( \angle NMO \) and \( \angle LOM \)) are alternate interior angles.