Question

given: △dfe is isosceles with base fe, (overline{fb}congoverline{ec}).
prove: △dfb ≅ △dec
complete the missing parts of the paragraph proof.
we know that triangle dfe is isosceles with base (overline{fe}) and that segment (overline{fb}) is congruent to segment (overline{ec}) because
segment (overline{df}) is congruent to segment (overline{}) by the definition of isosceles triangle.
since these segments are congruent, the base angles, angles (overline{}) are congruent by the isosceles triangle
theorem. therefore, triangles (overline{}) are congruent by sas.

Explanation:

Step1: Identify the congruent sides in the isosceles triangle

Since $\triangle DFE$ is isosceles with base $\overline{FE}$, by the definition of isosceles triangle, segment $DF$ is congruent to segment $DE$.

Step2: Find the congruent base angles

Since these segments are congruent, the base angles, angles $\angle DFB$, $\angle DEC$ are congruent by the isosceles triangle theorem.

Step3: Apply the SAS (Side - Angle - Side) congruence criterion

We know that segment $FB\cong EC$ (given), $\angle DFB\cong\angle DEC$ (from step 2), and $DF\cong DE$ (from step 1). So, triangles $\triangle DFB$, $\triangle DEC$ are congruent by SAS.

Answer:

DE, $\angle DFB$, $\angle DEC$, $\triangle DFB$, $\triangle DEC$