Question

given: \\( \triangle dfe \\) is isosceles with base \\( \overline{fe} \\); \\( \overline{fb} \cong \overline{ec} \\).
prove: \\( \triangle dfb \cong \triangle dec \\)

complete the missing parts of the paragraph proof.

we know that triangle dfe is isosceles with base \\( \overline{fe} \\) and that segment fb
is congruent to segment ec because

segments
segment by the definition of isosceles t
segments are congruent, the base angles, angles
congruent by the isosceles triangle theorem. therefore, triangles
are congruent by sas.

Explanation:

Step1: Given Information

We are given that \(\triangle DFE\) is isosceles with base \(\overline{FE}\) and \(\overline{FB}\cong\overline{EC}\). The first blank is filled with "it is given" as per the problem statement.

Step2: Isosceles Triangle Property

In an isosceles triangle \(\triangle DFE\) with base \(\overline{FE}\), the legs are congruent. So, \(\overline{DF}\cong\overline{DE}\) by the definition of an isosceles triangle.

Step3: Base Angles of Isosceles Triangle

For an isosceles triangle \(\triangle DFE\) with \(\overline{DF}\cong\overline{DE}\), the base angles \(\angle DFE\) and \(\angle DEF\) are congruent by the isosceles - triangle theorem.

Step4: SAS Congruence Criterion

We have \(\overline{DF}\cong\overline{DE}\) (from step 2), \(\angle DFE\cong\angle DEF\) (from step 3), and \(\overline{FB}\cong\overline{EC}\) (given). So, \(\triangle DFB\cong\triangle DEC\) by the Side - Angle - Side (SAS) congruence criterion.

Answer:

The first blank is filled with "it is given" (as per the problem's given conditions). The second blank is \(\overline{DF}\) (since in an isosceles triangle \(\triangle DFE\) with base \(\overline{FE}\), the non - base sides are congruent). The third blank is \(\angle DFE\) and \(\angle DEF\) (base angles of isosceles triangle \(\triangle DFE\)). The fourth blank is \(\triangle DFB\) and \(\triangle DEC\) (the triangles we are proving congruent by SAS).