Question

use the given information to prove that \\( \overline{fg} \parallel \overline{kh} \\).
you are not allowed to use the transitive property for parallel lines.
(you may use the transitive property for \=\ or \\cong\, though.)

given: \\( \overline{fe} \parallel \overline{gj} \\)
\\( \angle 6 \cong \angle 1 \\)
prove: \\( \overline{fg} \parallel \overline{kh} \\)

Explanation:

Step1: Identify Given Information

We are given \( \overline{FE} \parallel \overline{GJ} \) and \( \angle 6 \cong \angle 1 \). Also, we know \( \angle 6 \) and \( \angle 5 \) are same - side interior angles (since \( \overline{FE} \parallel \overline{GJ} \), by the Consecutive Interior Angles Theorem, \( \angle 6 + \angle 5 = 180^\circ \) if they were supplementary, but actually, since \( \overline{FE} \) and \( \overline{GJ} \) are parallel and \( \overline{FG} \) is a transversal, \( \angle 6\) and \( \angle 5\) are corresponding or alternate? Wait, no, \( \overline{FE}\) is vertical, \( \overline{FG}\) is horizontal, \( \overline{GJ}\) is vertical. So \( \overline{FE}\parallel\overline{GJ}\) (both vertical) and \( \overline{FG}\) is horizontal, so \( \angle 6\) and \( \angle 5\) are right angles? Wait, maybe better to use the given \( \angle 6\cong\angle 1\) and the fact that \( \overline{FE}\parallel\overline{GJ}\) implies \( \angle 6\cong\angle 5\) (corresponding angles, since \( \overline{FG}\) is transversal to \( \overline{FE}\) and \( \overline{GJ}\)). Then by transitive property (since \( \angle 6\cong\angle 1\) and \( \angle 6\cong\angle 5\)), \( \angle 5\cong\angle 1\). Then \( \angle 5\) and \( \angle 1\) are corresponding angles for lines \( \overline{FG}\) and \( \overline{KH}\) with transversal \( \overline{GJ}\). So if corresponding angles are congruent, then lines are parallel.

Wait, let's start step - by - step:

Step1: State Given Parallel Lines

Statement: \( \overline{FE} \parallel \overline{GJ} \)
Reason: Given

Step2: Identify Corresponding Angles

Since \( \overline{FE} \parallel \overline{GJ} \) and \( \overline{FG} \) is a transversal, \( \angle 6 \cong \angle 5 \) (Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent)

Step3: Use Transitive Property of Congruence

Statement: \( \angle 5 \cong \angle 1 \)
Reason: Transitive Property of Congruence (since \( \angle 6 \cong \angle 1 \) (given) and \( \angle 6 \cong \angle 5 \) (from step 2))

Step4: Prove Lines are Parallel

Statement: \( \overline{FG} \parallel \overline{KH} \)
Reason: Corresponding Angles Converse (If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. Here, \( \angle 5 \) and \( \angle 1 \) are corresponding angles with transversal \( \overline{GJ} \) for lines \( \overline{FG} \) and \( \overline{KH} \))

Answer:

To prove \( \overline{FG}\parallel\overline{KH} \):

1. \( \overline{FE} \parallel \overline{GJ} \) (Given)
2. \( \angle 6 \cong \angle 5 \) (Corresponding Angles Postulate, as \( \overline{FE}\parallel\overline{GJ}\) and \( \overline{FG}\) is transversal)
3. \( \angle 6 \cong \angle 1 \) (Given)
4. \( \angle 5 \cong \angle 1 \) (Transitive Property of Congruence, from steps 2 and 3)
5. \( \overline{FG} \parallel \overline{KH} \) (Corresponding Angles Converse, since \( \angle 5\cong\angle 1\) and they are corresponding angles for \( \overline{FG}\) and \( \overline{KH}\) with transversal \( \overline{GJ}\))

The final answer is that we have proven \( \overline{FG}\parallel\overline{KH}\) using the given information, corresponding angles postulate, transitive property of congruence, and corresponding angles converse.