Question

given: ( angle e cong angle g ), ( overline{de} parallel overline{fg} ) prove: ( overline{ef} parallel overline{gh} ) 1. ( angle e cong angle g ), ( overline{de} parallel overline{fg} ) 2. click here to insert 2. if two parallel lines are cut by a transversal, then their alternate interior angles are congruent.

Explanation:

Step1: Identify the transversal and alternate interior angles

Since \( \overline{DE} \parallel \overline{FG} \) and \( \overline{EF} \) is a transversal, by the Alternate Interior Angles Theorem, \( \angle E \cong \angle EFG \).

Step2: Use the given congruence

We know \( \angle E \cong \angle G \) (given). By the Transitive Property of Congruence, \( \angle EFG \cong \angle G \).

Step3: Apply the Alternate Interior Angles Converse

Since \( \angle EFG \cong \angle G \) and \( \overline{FG} \) is a transversal for \( \overline{EF} \) and \( \overline{GH} \), by the Converse of the Alternate Interior Angles Theorem, \( \overline{EF} \parallel \overline{GH} \).

Answer:

To prove \( \overline{EF} \parallel \overline{GH} \):

1. Given: \( \angle E \cong \angle G \), \( \overline{DE} \parallel \overline{FG} \)
2. Alternate Interior Angles Theorem: Since \( \overline{DE} \parallel \overline{FG} \) and \( \overline{EF} \) is a transversal, \( \angle E \cong \angle EFG \).
3. Transitive Property: From \( \angle E \cong \angle G \) (given) and \( \angle E \cong \angle EFG \) (step 2), we get \( \angle EFG \cong \angle G \).
4. Converse of Alternate Interior Angles Theorem: Since \( \angle EFG \cong \angle G \) (alternate interior angles) and \( \overline{FG} \) is a transversal for \( \overline{EF} \) and \( \overline{GH} \), \( \overline{EF} \parallel \overline{GH} \).

Thus, \( \boldsymbol{\overline{EF} \parallel \overline{GH}} \) is proven.