Question

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

Explanation:

Step1: Properties of Isosceles Trapezoid

In an isosceles trapezoid \(FGHJ\) with \(FJ\cong GH\), \(\angle FGH\cong\angle JHG\) (base - angles of an isosceles trapezoid are congruent) and \(FG = JH\) (legs of an isosceles trapezoid are congruent). Also, \(GH = FJ\) (given).

Step2: SAS Congruence Criterion

For \(\triangle FHG\) and \(\triangle GJF\):

• \(FG = JH\) (legs of isosceles trapezoid)
• \(\angle FGH=\angle JHG\) (base - angles of isosceles trapezoid)
• \(GH = FJ\) (given)

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle FHG\cong\triangle GJF\).

Step3: CPCTC

Since \(\triangle FHG\cong\triangle GJF\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, \(\overline{FH}\cong\overline{GJ}\).

Answer:

1. Reason: Side - Angle - Side (SAS) congruence criterion.
2. Statement: \(\overline{FH}\cong\overline{GJ}\); Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC).