Question

given: isosceles trapezoid efgh
prove: \\( \triangle fhe \cong \triangle geh \\)
it is given that trapezoid efgh is an isosceles
trapezoid. we know that \\( \overline{fe} \cong \overline{gh} \\) by the definition of
. the base angle theorem of
isosceles trapezoids verifies that angle is
congruent to angle. we also see that \\( \overline{eh} \cong \overline{eh} \\)
by the property. therefore, by,
we see that \\( \triangle fhe \cong \triangle geh \\).

Explanation:

• In an isosceles trapezoid, the non - parallel sides (legs) are congruent. So, for isosceles trapezoid \(EFGH\), \(\overline{FE}\cong\overline{GH}\) by the definition of an isosceles trapezoid.
• The base - angle theorem of isosceles trapezoids states that base angles are congruent. In trapezoid \(EFGH\) with bases \(FG\) and \(EH\), \(\angle FEH\cong\angle GHE\).
• \(\overline{EH}\cong\overline{EH}\) by the reflexive property (a segment is congruent to itself).
• To prove \(\triangle FHE\cong\triangle GEH\), we use the Side - Angle - Side (\(SAS\)) congruence criterion. We have two sides and the included angle congruent: \(\overline{FE}\cong\overline{GH}\), \(\angle FEH\cong\angle GHE\), and \(\overline{EH}\cong\overline{EH}\).

Answer:

• First blank: isosceles trapezoid
• Second blank: \(\angle FEH\)
• Third blank: \(\angle GHE\)
• Fourth blank: reflexive
• Fifth blank: \(SAS\) (Side - Angle - Side)