Question

1. given: $overleftrightarrow{mn} parallel overleftrightarrow{op}$

$overline{mn} cong overline{op}$
prove: $angle m cong angle p$

possible statements:
(a) segment $no$ ≅ segment $no$, (b) $angle m ≅ angle p$
(c) $angle mno ≅ angle pon$, (d) $angle mon ≅ angle pno$

possible reasons:
(e) given, (f) aas, (g) sas, (h) asa, (i) sss, (j) cpctc, (k) prove

reason #1: ____
statement #2: ____
statement #3:____
reason #4: ____
reason #5: ____

Explanation:

Step 1: Reason #1

The first statement is given as \(\overline{MN} \cong \overline{OP}\) and \(\overline{MN} \parallel \overline{OP}\), so the reason is "Given".
Reason #1: E (Given)

Step 2: Statement #2

Since \(\overline{MN} \parallel \overline{OP}\), the alternate interior angles \(\angle MNO\) and \(\angle PON\) are congruent. So the statement is \(\angle MNO \cong \angle PON\).
Statement #2: C (\(\angle MNO \cong \angle PON\))

Step 3: Statement #3

By the reflexive property, \(\overline{NO} \cong \overline{NO}\). So the statement is \(\overline{NO} \cong \overline{NO}\).
Statement #3: A (segment \(NO \cong\) segment \(NO\))

Step 4: Reason #4

We have \(\overline{MN} \cong \overline{OP}\), \(\angle MNO \cong \angle PON\), and \(\overline{NO} \cong \overline{NO}\). This satisfies the SAS (Side - Angle - Side) congruence criterion for triangles \(\triangle MNO\) and \(\triangle PON\). So the reason is "SAS".
Reason #4: G (SAS)

Step 5: Reason #5

Since \(\triangle MNO \cong \triangle PON\), the corresponding parts of congruent triangles are congruent (CPCTC). So \(\angle M \cong \angle P\) because they are corresponding parts of the congruent triangles.
Reason #5: J (CPCTC)

Answer:

Reason #1: E (Given)
Statement #2: C (\(\angle MNO \cong \angle PON\))
Statement #3: A (segment \(NO \cong\) segment \(NO\))
Reason #4: G (SAS)
Reason #5: J (CPCTC)