Question

given: \\( \overline { m n } \parallel \overline { p o } ; \overline { m p } \parallel \overline { n o } \\)
prove: \\( \delta m n p \cong \delta o p n \\)
you are given the information that
. since \\( \overline { n p } \\) is a transversal
cutting parallel segments, \\( \overline { m n } \\) and \\( \overline { p o } \\), you know that \\( \angle m n p \cong \angle o p n \\) because of the
theorem. also, since \\( \overline { n p } \\) is a transversal cutting parallel
segments, \\( \overline { m p } \\) and \\( \overline { n o } \\), you know that
for the same reason.
you also know that \\( \overline { n p } \cong \overline { n p } \\) because of the
property of
congruence. since you have two angles and an included side of \\( \delta m n p \\) congruent to the
corresponding two angles and included side of \\( \delta o p n \\), you can conclude that
because of asa.
a. \\( \overline { m n } \parallel \overline { p o } ; \overline { m p } \parallel \overline { n o } \\)
b. symmetric
c. \\( \delta m n p \cong \delta o p n \\)
d. alternate exterior angles
e. \\( \angle o n p \cong \angle m p n \\)
f. transitive
g. reflexive
h. alternate interior angles

Explanation:

Step1: Identify given parallel lines

The first blank is filled with the given information \( \overline{MN}\parallel\overline{PO};\overline{MP}\parallel\overline{NO}\) (option a).

Step2: Use alternate - interior angles theorem

When a transversal cuts two parallel lines, alternate - interior angles are congruent. For transversal \( \overline{NP}\) cutting \( \overline{MN}\parallel\overline{PO}\), \( \angle MNP\cong\angle OPN\) by the Alternate Interior Angles Theorem (option h).

Step3: Find another pair of congruent angles

For transversal \( \overline{NP}\) cutting \( \overline{MP}\parallel\overline{NO}\), \( \angle ONP\cong\angle MPN\) (option e).

Step4: Use reflexive property

\( \overline{NP}\cong\overline{NP}\) by the Reflexive Property of Congruence (option g).

Step5: Conclude triangle congruence

Since we have two angles and the included side of \( \triangle MNP\) congruent to the corresponding two angles and included side of \( \triangle OPN\), \( \triangle MNP\cong\triangle OPN\) (option c).

Answer:

a. \( \overline{MN}\parallel\overline{PO};\overline{MP}\parallel\overline{NO}\)
h. Alternate Interior Angles
e. \( \angle ONP\cong\angle MPN\)
g. Reflexive
c. \( \triangle MNP\cong\triangle OPN\)