Question

given: fghj is an isos - celes trapezoid with (overline{fj}congoverline{gh}). prove: (overline{fh}congoverline{gj})

Explanation:

Step1: Identify the base angles

Since \(FGHJ\) is an isosceles trapezoid, \(\angle FGH\cong\angle JHG\) (by the property that the base angles of an isosceles trapezoid are congruent).

Step2: Use the Side - Angle - Side (SAS) congruence criterion

We know that \(FG = JH\) (given \(FGHJ\) is an isosceles trapezoid with \(FJ\cong GH\), and in an isosceles trapezoid the non - parallel sides are equal), \(\angle FGH\cong\angle JHG\) (from Step 1), and \(GH = JH\) (given \(FJ\cong GH\) and \(FG = JH\)).
So, \(\triangle FGH\cong\triangle JHG\) (by SAS: \(FG = JH\), \(\angle FGH=\angle JHG\), \(GH = JH\)).

Step3: Conclude the congruence of the diagonals

Since \(\triangle FGH\cong\triangle JHG\), then \(\overline{FH}\cong\overline{GJ}\) (corresponding parts of congruent triangles are congruent).

Answer:

1. \(\angle FGH\cong\angle JHG\)
2. \(\triangle FGH\cong\triangle JHG\); SAS (Side - Angle - Side) congruence criterion