Question

proving parallel lines with alternate exterior angles

given: \\(\angle 1 \cong \angle 2\\)
prove: \\(p \parallel q\\)

angles lines statements reasons

transitive property
vertical angles theorem
def. of vertical \\(\angle\\)s
converse of corresponding angles theorem

statements

1. \\(\angle 1 \cong \angle 2\\)

reasons

1. given

Explanation:

Identify the geometric relationships

We are given a geometric proof with:

• Given: \(\angle 1 \cong \angle 2\)
• Prove: \(p \parallel q\)

Looking at the diagram:

• \(\angle 1\) and \(\angle 3\) are vertical angles formed by the intersection of line \(p\) and the transversal.
• Therefore, \(\angle 1 \cong \angle 3\) by the Vertical Angles Theorem.

Determine the next logical step

The first statement is:

1. \(\angle 1 \cong \angle 2\) (Reason: Given)

To connect \(\angle 2\) and \(\angle 3\), we first establish the relationship between \(\angle 1\) and \(\angle 3\):

1. \(\angle 1 \cong \angle 3\) (Reason: vertical angles theorem)

Apply the transitive property

Once we have \(\angle 1 \cong \angle 2\) and \(\angle 1 \cong \angle 3\), we can conclude:

1. \(\angle 3 \cong \angle 2\) (Reason: transitive property)

Complete the parallel lines proof

Since \(\angle 3\) and \(\angle 2\) are corresponding angles and they are congruent, we can prove:

1. \(p \parallel q\) (Reason: converse of corresponding angles theorem)

Formulate the requested next statement

The prompt asks to "Assemble the next statement" (Statement 2).
The next logical statement is:

• Statement: \(\angle 1 \cong \angle 3\)
• Reason: vertical angles theorem

In the interactive interface, the statement boxes are split into three slots for the statement components:

• Left slot: \(\angle 1\)
• Middle slot: \(\cong\)
• Right slot: \(\angle 3\)

And the corresponding reason is:

• vertical angles theorem

Answer:

The next statement and reason to assemble in the proof are:

Statements: <blank>\(\angle 1 \cong \angle 3\)</blank>
Reasons: <blank>vertical angles theorem</blank>