Question

given: \\( \overline { e f } \parallel \overline { g h } \\) and \\( \overline { a b } \parallel \overline { c d } \\).
prove: \\( \angle g j i \cong \angle l k f \\).
note: quadrilateral properties are not permitted in this
proof.

note: \\( \overline { a b } \\), \\( \overline { c d } \\), \\( \overline { e f } \\) and \\( \overline { g h } \\) are segments.

Explanation:

Step1: Corresponding angles of parallel lines

Since \(\overline{EF}\parallel\overline{GH}\), \(\angle GJI\) and \(\angle BIF\) are corresponding angles, so \(\angle GJI=\angle BIF\) (Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent).

Step2: Corresponding angles of parallel lines

Since \(\overline{AB}\parallel\overline{CD}\), \(\angle BIF\) and \(\angle LKF\) are corresponding angles, so \(\angle BIF = \angle LKF\) (Corresponding Angles Postulate).

Step3: Transitive property of congruence

By the transitive property of congruence (If \(a = b\) and \(b=c\), then \(a = c\)), since \(\angle GJI=\angle BIF\) and \(\angle BIF=\angle LKF\), we have \(\angle GJI\cong\angle LKF\).

Answer:

\(\angle GJI\cong\angle LKF\) is proved as above.