Question

given: hf || jk; hg ≅ jg prove: △fhg ≅ △kjg to prove that the triangles are congruent by asa, which statement and reason could be used as part of the proof? ∠fgh ≅ ∠kgj because vertical angles are congruent. ∠jkg ≅ ∠hfg because vertical angles are congruent. ∠fhg ≅ ∠jkg because right angles are congruent. ∠hfg ≅ ∠kjg because alternate interior angles are congruent.

Explanation:

• For proving \(\triangle FHG\cong\triangle KJG\) by \(ASA\) (Angle - Side - Angle), we need two angles and the included side to be congruent.
• We know \(HG\cong JG\) (given).
• Since \(HF\parallel JK\), by the alternate - interior angles theorem (when two parallel lines are cut by a transversal, alternate - interior angles are congruent), \(\angle HFG\) and \(\angle KJG\) are alternate - interior angles (with transversal \(FG\) for parallel lines \(HF\) and \(JK\)).
• Also, \(\angle FGH\) and \(\angle KGJ\) are vertical angles (by the definition of vertical angles, which are congruent).

Answer:

\(\angle FGH\cong\angle KGJ\) because vertical angles are congruent.