Question

given: (overline{gh}paralleloverline{ij}), (overline{gh}congoverline{ij}), (overline{gk}congoverline{il})
prove: (angle ihkcongangle gjl)

1. (angle hkicongangle jlg)
2. if two angles are sup - plementary to congru - ent angles, then they are congruent.
3. (\triangle hkicong\triangle jlg)
4. side - angle - side
5. click here to insert
6. click here to insert

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle HKI\cong\triangle JLG\) (from statement 12), then \(HI = GJ\) (corresponding parts of congruent triangles are congruent).

Step2: Use the property of parallel lines

Given \( \overline{GH}\parallel\overline{IJ}\), so \(\angle GHI=\angle JIH\) (alternate - interior angles).

Step3: Use the Side - Angle - Side (SAS) criterion

We know \( \overline{GH}\cong\overline{IJ}\) (given), \(\angle GHI=\angle JIH\) (from step 2), and \(HI = GJ\) (from step 1). So, \(\triangle IHK\cong\triangle GJL\) by SAS.

Step4: Conclude the angle congruence

Since \(\triangle IHK\cong\triangle GJL\), then \(\angle IHK\cong\angle GJL\) (corresponding parts of congruent triangles are congruent).

Answer:

\(\angle IHK\cong\angle GJL\) because \(\triangle IHK\cong\triangle GJL\) (by SAS, as \( \overline{GH}\cong\overline{IJ}\), \(\angle GHI=\angle JIH\), and \(HI = GJ\)) and corresponding parts of congruent triangles are congruent.