Question

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

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

Step2: Alternate exterior angles

Since \(\overline{AB}\parallel\overline{CD}\), \(\angle BIK\) and \(\angle LKF\) are alternate exterior angles, so \(\angle BIK = \angle LKF\) (Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternate exterior angles are congruent).

Step3: Transitive property

By the transitive property of equality (If \(a = b\) and \(b=c\), then \(a = c\)), since \(\angle GJI=\angle BIK\) and \(\angle BIK=\angle LKF\), we have \(\angle GJI=\angle LKF\). So \(\angle GJI\cong\angle LKF\) (If two angles have the same measure, they are congruent).

Answer:

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