Question

given: ( overline{gh} parallel overline{ij} ), ( overline{gh} cong overline{ij} ), ( overline{gk} cong overline{il} ) prove: ( angle ihk cong angle gjl )

Explanation:

Step1: Identify the given information

Given \( \overline{GH}\parallel\overline{IJ}\), \( \overline{GH}\cong\overline{IJ}\), \( \overline{GK}\cong\overline{IL}\)

Step2: Use the property of parallel lines

Since \( \overline{GH}\parallel\overline{IJ}\), by the Alternate Interior Angles Theorem, \( \angle HGK\cong\angle JIL\)

Step3: Prove the triangles congruent

We have \( \overline{GH}\cong\overline{IJ}\), \( \angle HGK\cong\angle JIL\), \( \overline{GK}\cong\overline{IL}\). So, by the Side - Angle - Side (SAS) congruence criterion, \( \triangle HGK\cong\triangle JIL\)

Step4: Use the property of congruent triangles

If \( \triangle HGK\cong\triangle JIL\), then \( \angle GKH\cong\angle ILJ\). Also, \( \angle IHK\) and \( \angle GJL\) can be shown to be congruent by considering the sum of angles in the figure. Another way is to note that after proving \( \triangle HGK\cong\triangle JIL\), we can use the fact that \( \angle IHK\) and \( \angle GJL\) are angles that can be related through the congruent triangles and the parallel lines.

Answer:

1. \( \overline{GH}\parallel\overline{IJ},\overline{GH}\cong\overline{IJ},\overline{GK}\cong\overline{IL}\)
2. Alternate Interior Angles Theorem (because \( \overline{GH}\parallel\overline{IJ}\))
3. \( \triangle HGK\cong\triangle JIL\) (by SAS)